# Combinatorial Solid Geometry, Boundary Representations, and Non-Manifold Geometry

Michael John Muuss and Lee A. Butler

Published in State of the Art in Computer Graphics: Visualization and Modeling D. F. Rogers, R. A. Earnshaw editors, Springer-Verlag, New York, 1991, pages 185-223.


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{\chaptertitlefont \noindent Combinatorial Solid Geometry,}
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\leftline{\hyphenpenalty=10000
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{\chaptertitlefont \noindent Boundary Representations,}
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{\chaptertitlefont \noindent and Non-Manifold Geometry}
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\author {Michael John Muuss and Lee A. Butler}
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\beginabstract

The history of solid modeling is reviewed. Both traditional combinatorial
solid geometry (CSG) systems and traditional boundary representation (B-rep)
solid modeling systems are considered.
CSG models are formed by the boolean combination of primitive'' solids.
Only the unevaluated CSG tree is stored,
rather than an explicit representation of the final surfaces.
Many
applications would like an {\it approximation} of these complex CSG shapes
expressed as a collection of planar N-gons.
This paper focuses on a
technique for computing approximate three dimensional surface
tessellations.
So far, B-rep solid modelers have used variations on the winged-edge data
structure. The winged-edge data structure can only describe manifold shapes,
limiting these systems to manifold geometry. While not restricting the set of
manufacturable parts, this has made implementing boolean operations on these
B-rep models more difficult.

Boolean operations and other useful modeling operations can be much more
easily implemented using a representation that permits non-manifold topology
to be expressed directly. A detailed look is taken at the radial-edge data
structure and how it represents non-manifold conditions.
While several previous publications describe the general nature of algorithms
required to perform boolean operations on non-manifold geometry, details are
sparse. This report describes the necessary algorithms in detail.

Some of the systems analysis issues that can result from integrating
non-manifold B-rep geometry into a hybrid CSG solid modeling system are
discussed.
Of particular note are
the specification of tessellation tolerancing,
the implementation of robust tessellation,
the difficulties of carrying dual representations of objects,
ray-tracing non-manifold B-rep solids,
using the B-rep for driving high-performance polygon rendering hardware,
and interfacing to facet-based analysis codes.
\endabstract

{
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To appear in
ed. Rogers and Earnshaw, Springer-Verlag.
}
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\def\rhead{Combinatorial Solid Geometry, B-Reps, and Non-Manifold Geometry}
\def\lhead{Michael John Muuss and Lee A. Butler}
The digital computer provides the opportunity for nearly infinite
variety in the representation of information within it.
The design of complex, expensive objects
is always an area ripe for technological improvements,
and with the emergence of graphics displays, computer aided
design (CAD) packages began to appear.
Designers of early CAD packages focused their efforts on the most
tedious, time-consuming, and unrewarding aspect of conventional design:
the process of converting a designer's sketches and notes into
finished engineering drawings -- the drafting process.
Only modest gains in efficiency were possible in creating
the first version of a design,
but tremendous gains were realized when design modifications were needed,
because the computer can retain and redraw the unchanged portions of
the drawing.

Once a computer aided drafting system has been used to create a
computer representation of a design,
the designer (and his management)
is often tempted to expect more out of the computer
system than simple drawings.  One would expect that a representation
that depicted the boundaries of an object from several views
would provide sufficient information for computing a whole
variety of useful facts about the model, such as center of mass,
volume, cross-sectional area, etc.
Two-dimensional drafting systems were not designed for this
sort of interrogation.

Four major types of difficulties have plagued wireframe systems.
First, the user is required to supply a large amount of information,
often at a very low level.
Because of the drafting heritage, some of
this information may be construction lines'' that do not actually
contribute to the ultimate shape of any object.
Second, because the user is providing such low-level information,
it is easy to define objects
which cannot be physically realized due to
non-closed faces and dangling lines.
Third, it is possible to construct a wireframe which
has ambiguous interpretations from different views.
Finally, wireframe models may include view-specific
lines representing false edges such as profile lines and silhouettes [REQU82].
Engineering drawings are suitable only for interpretation by human
beings, not for automatic computerized analysis.

As a new product is being designed, there are generally two or more
different kinds of engineering analyses that need to be performed,
such as
structural analysis,
thermal analysis,
computational fluid dynamics (CFD), and
vulnerability analysis,
as well as the calculation of predictive
optical, radar, infra-red (IR), and X-ray images or signatures''.
Therefore, computerized drawings cannot be used
to provide the geometric input required for these analysis codes.
These requirements have focused attention on the evolution
of an entirely different approach to representing objects:  solid models.

A solid model [MUUS87a] is
a computer description of closed, solid, three dimensional shapes
represented by  an analytical framework within which the three dimensional
material can be completely and unambiguously defined [DEIT83].
Solid models differ from drafting-type systems in several
important ways:
objects are composed of combinations of primitive objects (some quite
complex), each of which is complete, unambiguous,
physically realizable, and modifiable.
Because these properties hold for the primitive objects, they hold
for any boolean combinations of them as well.

Completeness is assured because the
representation contains a full description of a piece of solid matter;
there is no view-specific information.
Because the solid matter is completely described, there
is no possibility for ambiguity.
For primitive solids defined by specifying parameters to an analytic
function, there is no possibility of having missing faces, loose edges, or
other similar defects.
Systems which offer boundary representations as primitive solids
must carefully validate each such solid when it is created.
A solid model is
always amenable to further modification by boolean combination with
other shapes.

These properties guarantee that
all the spatial information necessary for any
subsequent analysis is directly available from the model representation.
Object structure and material properties can be computed
at any arbitrary point in the model
at any time.
Therefore, solid modeling technology is particularly suited to the
automation of many manufacturing and analysis tasks.

%% was 4.5 by 3.25
\captionfig 3.5in by 2.5in, design-loop, Figure 1 -- The Design Loop.

\eject
Solid models are very useful for generating drawings or pictures of
the modeled object from any viewpoint.
This capability alone usually pays for the cost of
developing the model.
However, the solid model has a much larger role in the design process
than simply automating the production of pictures and engineering
drawings.
Properly utilized, the solid model becomes the central element in the
iterative process of taking a design from idea to prototype design
to working design to optimized design.
The model can be subjected to
numerous engineering analyses, allowing
the effects of varying many parameters
to be studied in a controlled and automatic way.
This iterative process is termed the design loop'', and is illustrated
in Figure 1.

In a full scale solid modeling system, there is no need for initial
drawings: the designer expresses the initial structures directly
into the modeling system's editor,
just as a modern author creates his rough draft''
directly into a word processor.
At the completion of each version of the design, the model is
subjected to a battery of analyses appropriate to the function of the
object being designed.
Strength, volume, weight, level of protection, and other similar evaluations
can be reported, along with the
production of a variety of images and/or drawings.
These automated analyses help identify weaknesses or deficiencies in
a design {\it early in the design process}.
By detecting flaws early, the designer has the
opportunity to correct his plans before having invested too much time
in a bad design, or the designer can
switch to an entirely different approach which may seem more promising
than the original one.

In this way, the solid modeling system allows the designer to
concentrate on the important, creative aspects of the design process.
Freeing the designer of routine analysis permits designs to be
finished in less time than previously required, or allows much more
rigorously optimized designs to be delivered in comparable
timeframes and at the same cost as unoptimized designs created using
older techniques [DEIT85].
Furthermore, the modeling system allows sweeping design changes to be
made quickly and cheaply, allowing great flexibility in the face of
ever changing requirements and markets.
The time needed to create a new product can be further decreased by
re-utilizing elements of earlier models and then modifying them as
appropriate.
If an existing component already in inventory is entirely suitable
for use in a new design, significant manufacturing and inventory
savings will be realized.
A highly interactive modeling system can allow full designs to be
completed in a matter of days, where weeks or months may have
previously been required [DEIT82].

Thus, the real payoff from building a solid geometric model
comes when it is time to analyze it.
This capability is so powerful that it ordinarily justifies any
extra time or equipment investments needed to support the
construction of the three dimensional solid model.
Allowing the designer the opportunity to explore and
analyze more design options will allow the development of the
highest quality product, while also improving the work environment of
the designer by eliminating boring, repetitive tasks.

Two major families of solid model representations exist,
The first representation, developed by MAGI under contract to
the Ballistic Research Lab (BRL)
in the early 1960s [MAGI67],
is the combinatorial solid geometry representation (CSG-rep).
Solid models of this type are expressed as boolean combinations of
primitive solids.
Each primitive solid is a geometric entity described by some set of
parameters that occupies a fixed volume in space.

The simplest solid that can be used is the halfspace [REQU82],
defined by the infinite plane $a x + b y + c z + d = 0$
plus all points on one side of that plane.
Systems which defined all objects in terms of
Boolean combinations of halfspaces
include SHAPES [LANI79]
and TIPS-1 [OKIN78].
While this choice of representation
limits these systems to modeling convex objects with
planar faces, and excludes smooth objects (or forces them to be
approximated), the simplicity of this representation lends itself to
very natural processing by VLSI hardware [KEDE85a, KEDE85b].
Most CSG-rep systems in use today
offer quite a variety of primitive solids, ranging from various types
of spheres and ellipsoids, boxes and cones,
and solids defined by swept or extruded curves.

The alternative to describing solids with primitives is to adopt a
boundary representation (B-rep),
of which there are two sub-types:
the {\it explicit} and {\it implicit} boundary representations.
In an explicit boundary representation, each solid is described
by an explicit specification of all the points on the surface of the
solid, typically by exhaustively listing the vertices of many planar
facets.
Alternatively, there are implicit boundary representations,
where the surface of the solid is described by an analytic function
such as Coons patches [COON67],
Bezier patches [BEZI74],
splines [deBO78],
etc.

Boundary representations offer the advantage of being able to
naturally model solid objects with arbitrarily shaped surfaces, but
can require a large amount of information to achieve acceptable
results.
Both CSG-reps and B-reps have certain advantages.
With only the traditional CSG primitives,
it can be exceedingly difficult and non-intuitive to
attempt to describe sculptured, free-form surfaces
as a boolean combination of primitives.
But similarly,
implementing powerful modeling operations like boolean intersection and
boolean differences {\it on the fundamental representation itself}
can be difficult with pure boundary representations.
Many current B-rep modelers implement boolean operations as an
external post-processing operation, because current schemes to
evaluate Boolean operations are not closed.
As an example of this,
B-spline $\cap$ B-spline might result in polygons
rather than another B-spline.
In a boundary representation closed under the set of boolean operations,
B-rep $\cap$ B-rep $\rightarrow$ B-rep.
Thus, pure B-rep systems may be difficult to use for some types of
objects, especially those with sculptured surfaces pierced by sharp
rectangular gouges [THOM84].

Even though existing systems are often considered as being purely
CSG-rep or B-rep, the reality is that many
production CAD systems are actually hybrids of the two approaches,
offering the designer the choice of primitive solids or boundary
representations, as appropriate for each task.
In practice, the implementation of the
CSG-rep and B-rep portions of
the software may be quite different, but at the highest level of abstraction
each representation is just a different way of viewing the other.
Faceted primitives such as boxes and wedges can be thought of as
explicit B-reps, and smooth primitives such as spheres and cones can
be thought of as implicit B-reps defined by analytic
functions.
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For more than thirty years,
solid geometric modeling methods have been used
to support engineering design and analyses [MAGI67, DEIT82, \break DEIT84a].
In such item-level studies
geometry and material information are passed to various
application codes to derive certain measures-of-performance.
This is commonly done in
structural analysis,
thermal analysis,
computational fluid dynamics (CFD), and
vulnerability analysis.
the techniqsues have been extended
to support many
predictive signature
models [DEIT84b]
including optical,
millimeter wave (MMW),
infra-red (IR), magnetic and X-ray signature generation tools.
It is important to note that this type of predictive analysis
must generally be supported by a solid geometric model.

The objective of a given application will to a large degree determine the
most natural'' form in which the model might be presented.
For example, a representation of just the {\it edges} of the objects in a model
would be suitable for a program attempting to construct a wire-frame display
of the model.
There exists another family of applications which must be able to find
the intersection of small object paths (e.g., photons) with the
model.
Generally, these alternatives are motivated by the representation of
a physical process being simulated, and each alternative
is useful for a whole family of applications.

\captionfig 4.5in by 3in, appl-interf, Figure 2 -- The Applications Interface.

Unfortunately,
it is not often the case that building only {\it one}
three dimensional model of
the product is enough.
Each of the different engineering
analysis software packages needed to perform the analyses
usually requires a different form of input.
As a result, more than one kind of geometric model may have
to be constructed.
However, rarely do these application
the three-dimensional geometry and material data base directly.
Rather, each application has a specific
interrogation method that is invoked to obtain
geometric and material attributes
from a source or reference file.
The physical simulation techniques used in the application software
are therefore constrained by the available techniques for
extracting geometric information from the model.
Each analysis package often requires a unique
form of input.
Without a central geometry database that can drive all
the analysis packages,
the designer can be forced into having to create
many different representations of each design,
one for each distinctly different type of analysis code.
This can be very costly and time consuming.
The time needed to create a
single model ranges between one man-week and several man-years,
depending on the complexity of the design.
Having to spend the effort to manually create the same design
in different formats to drive several analysis codes
is an unfortunate and expensive necessity.

has been to develop a broad set of
analysis codes which access the same geometry database [DEIT84b].
These analyses cover the spectrum from engineering decision aids,
to design validators, to signature prediction codes, to
the generation of wireframe drawings,
to high-resolution image generation
for management comprehension and sales advantage.
Key analysis capabilities have been developed to assess the
strength, weight, protection, and performance levels offered by
the structures represented by a solid model.
Using this analysis information and additional domain-specific
applications tools makes it possible to produce highly detailed
designs
constructed with a philosophy of {\it system optimization}
right from the start [DEIT88].
This facilitates the rapid
development of products with the desired levels
of performance at the best attainable price.

To accomplish all these goals,
provides a variety of procedural interfaces so that the
diverse collection of analysis codes can be driven from a single, central
geometric model [MUUS90b].
These procedural interfaces follow the natural object-oriented
programming interface.
An application program retrieves one or more objects from the model
database, and then requests those objects to either interrogate themselves
in the desired way, or to convert themselves into
the desired representation.
This applications interface is depicted in Figure 2.
The software
is designed to be highly portable to different hardware architectures,
platforms and environments,
and is implemented
using the C programming language [RITC78b]
running on the ${\rm UNIX^{tm}}$ operating system [RITC78a].

The interactive model editor {\bf mged} program employs
three dimensional wireframe outlines of the various solid objects
in order to maintain the highest possible speed of user interaction.
In addition to the primary use in supporting interactive geometry
viewing and modification,
wireframes are also useful for visualizing complex analysis results.
A particularly powerful form of this is to create a color display
of the output of an analysis code (for example, temperature
distribution across the surface of an object),
and then overlay the analysis data with a wireframe drawing
of the geometry.
Wireframes are also very useful for the previewing of animation
sequences.
The conversion of database objects into wireframe drawings
is the simplest of the application interfaces,
and is very easy for the application program to utilize.

After the user specifies which objects from the model database should
be displayed, {\bf mged} retrieves the necessary database records
and invokes the {\bf ft_plot()} interface provided by {\bf librt}.
{\bf ft_plot()} passes the database object
to the appropriate object-specific wireframe
converter, which generates a wireframe outline of that object.
The wireframe is composed of a collection of
three dimensional virtual pen-plotter
{\it move} and {\it draw} operations, returned to the application
as a linked list of {\bf vlist} structures attached to the application
Each {\bf vlist} structure has three elements,
vl_pnt, the XYZ coordinates of a point in space,
vl_draw, a flag which indicates whether the virtual pen should
be moved invisibly from the current position to
vl_pnt (vl_draw= VL_CMD_LINE_MOVE) or
moved visibly,
drawing a line from the current position to
vl_pnt (vl_draw= VL_CMD_LINE_DRAW).

Many phenomena that are ordinarily difficult to model
can be handled simply and elegantly with ray-tracing.
For example,
an illumination model
based on ray-tracing first fires a ray from the eye plane into
the model geometry.
To approximate the total amount of light at that point in the model,
the algorithm simply fires a ray at each light source
and sums the contributions from visible light sources.
Ray-tracing also makes it
easy to deal with objects that are partly or entirely reflective,
and with transparent objects that have varying refractive indices.
Furthermore, by applying the proper sorts of dither [COOK84],
and other effects are
easily achieved.

The power of the lighting model
can be further extended by making a provision to record the paths of all
the rays followed when computing the light intensity for each pixel
in an auxiliary file.
This capability
allows one to follow the path of the light rays passing through lenses
reflecting from mirrors while performing image rendering,
Studying the paths of light rays as they are repeatedly bent
by passing from air to glass and back again has traditionally
been a painstaking manual procedure for lens designers.
By modeling,
it becomes possible to predict lens behavior, including making a
determination of the exact focal length, finding the precise influence of
spherical distortions and edge effects, determining the amount of
image distortion due to internal reflection and scattering,
and finding the level of reflections from the lens mounting hardware.
Furthermore, experiments can be conducted to
determine the effects of adding or removing
baffles, irises, special lens coatings, etc.

Rays begin at a point $\vec P$,
and proceed infinitely in a given direction $\vec D$.
The {\it direction vector} or {\it direction cosines} for the ray
$( \vec D_x , \vec D_y , \vec D_z )$
are the cosines of
the angle between the ray and each of the Cartesian axes.
This vector $\vec D$ is of unit length, i.e. $| \vec D | = 1$.
Any point $\vec A$ on a ray may be expressed as a linear
combination of $\vec P$ and $\vec D$ by the formula
$\vec A ~=~ \vec P ~+~ t ~*~ \vec D$
where valid values for $t$ are in the range $[ ~0,~ \infty )$.

The traditional approach to ray-tracing has been batch-oriented,
with the user defining a set of viewing angles'',
initiating a large batch job to compute all the ray intersections,
and then post-processing all the ray data into some meaningful form.
However, the major drawback of this approach is that the application
has no immediate control over ray paths, making another batch run
necessary for each level of reflection, etc.

In order to be successful, applications need: (1) interactive control of ray
paths, to naturally implement reflection, refraction, and fragmenting into
multiple subsidiary rays, and (2) the ability to fire rays in arbitrary
directions from arbitrary points.
Nearly all non-batch implementations have closely
coupled a specific application (typically a model of illumination) with the
ray-tracing code, allowing efficient and effective control of the ray paths.
The most flexible approach of all is to provide the ray-tracing
capability through a general-purpose library, and make the functionality
available to any application as needed.
For example, the decision of when a ray should be reflected,
transmitted, or absorbed should be entirely under the control
of the application program.

The third generation ray-tracing capability in the BRL-CAD Package
is a set of library routines in {\bf librt}
to allow application programs to
intersect rays with model geometry.
There are two parts to the
interface: preparation'' routines and the actual ray-tracing routine.
{\bf rt_dirbuild()} opens the database file, and builds the
{\bf rt_gettree()}
adds a database sub-tree to the active model space,
and can be called multiple times
to join different parts of the database together.

To compute the intersection of a ray with the geometry in the
active model space, the application must call {\bf rt_shootray()}
once for each ray.
Ray behaviors such as
perspective, reflection, refraction, etc,
are entirely determined by the applications program logic,
and not by the ray-tracing library.
The ray-path specification
determined by the applications program is passed as a parameter
to {\bf rt_shootray()} in the  {\bf application} structure,
which contains five major elements:
the vector a_ray.r_pt ($\vec P$) which is
the starting point of the ray,
the vector a_ray.r_dir ($\vec D$) which is
the unit-length direction vector,
the pointer *a_hit() to an application-provided
routine to be called when some geometry is hit by the ray,
the pointer *a_miss() to an application-provided
routine to be called when the ray does not hit any geometry,
and the variable a_onehit.
various locations for applications to store state information
such as recursion level, intermediate color values,
and cumulative ray distance.

When the a_onehit variable is set to zero,
the ray is traced through the entire model.
Applications such as lighting models may often only be interested
in the first object hit;  in this case, a_onehit may be
set to the value one to stop ray-tracing
as soon as the ray has intersected at least one piece of geometry.
Similarly, if only the first three hits are required
(such as in the routine that refracts light through glass),
then a_onehit may be given the value of three.
Then, at most three hit points will be returned,
an in-hit, an out-hit, and a subsequent in-hit.
When only a limited number of intersections are required,
the use of this flag can provide a significant savings in run-time.

The {\bf rt_shootray()} function is designed for full recursion so that
the application provided a_hit()/a_miss() routines can themselves
fire additional rays by recursively calling {\bf rt_shootray()}
before deciding their own return value.
In addition, the function {\bf rt_shootray()} is fully capable of operating
in parallel with other instances of itself in the same address space,
allowing the application to take advantage of parallel hardware capabilities
where such exist.

A simple application program that fires one ray at a model and prints
the result is included below, to demonstrate the simplicity of the
interface to {\bf librt}.

\vbox{\beginverbatim
struct application ap;
struct rt_i *rtip;
main() {
rtip = rt_dirbuild(model.g'');
rt_gettree(rtip, car'');
rt_prep(rtip);
VSET( ap.a_point, 100, 0, 0 );
VSET( ap.a_dir, -1, 0, 0 );
ap.a_hit = &hit_geom;
ap.a_miss = &miss_geom;
ap.a_rt_i = rtip;
rt_shootray( &ap );
}
\endverbatim }
\vbox{\beginverbatim
hit_geom(app, pp)
struct application *app;
struct partition *pp;
{
printf(Hit %s'', pp->pt_forw->pt_regionp->reg_name);
}
miss_geom(){
printf(Missed'');
}
\endverbatim }

If a given ray hits something,
the a_hit() routine is called, and is provided
a pointer to the head of a doubly-linked list of {\bf partition}
structures.
Each {\bf partition} structure contains information about
a line segment along the ray;  the partition has both an in'' ({\bf pt_inhit})
and an out'' ({\bf pt_outhit}) hit point.
Each hit point is characterized by the hit distance {\bf hit_dist},
which is the distance $t$ from the starting point {\bf r_pt}
along the ray to the hit point.
The linked list of {\bf partition} structures is sorted by ascending
values of {\bf hit_dist}.
As a result of this definition, the line-of-sight'' distance between
any two hit points can be determined simply by subtracting the
two {\bf hit_dist} values.
This will give the distance between the hit points, in millimeters.

If the variable {\bf a_onehit} was set non-zero, then only the first {\bf a_onehit}
hit points along the partition list are guaranteed to be correct;
any additional hit points provided should be ignored.
This is usually important only when {\bf a_onehit} was set to an odd number;
the value of {\bf pt_outhit} in the last {\bf partition} structure may
not be accurate, and should be ignored.

If the actual three-space coordinates of the hit point are required,
they can be computed into the {\bf hit_point} element with
the C-language version of $\vec A = \vec P + t * \vec D$:
\beginverbatim
VJOIN1(hp->hit_point, rayp->r_pt, hp->hit_dist, rayp->r_dir);
\endverbatim

As an efficiency measure, only the hit distances are computed when
a ray is intersected with the model geometry.
The surface normal at any hit point can be easily
acquired by executing a C macro.
In addition to providing the unit-length outward-pointing surface
normal in struct {\bf hit} element {\bf hit_normal},
this macro also computes
the three-space coordinates of
the hit point in struct {\bf hit} element {\bf hit_point}:
\beginverbatim
RT_HIT_NORM( hitp, stp, rayp );
\endverbatim

For any hit point, after the surface normal has been computed,
the Gaussian surface curvature [STOK69] at that hit point can be acquired
by executing the C macro:
\beginverbatim
RT_CURVE( curvp, hitp, stp );
\endverbatim

At the hit point, there exists exactly one pair of orthogonal
directions also orthogonal to the surface normal $\vec N$
for which the values of $c$ take on
the minimum and maximum values $c_1$ and $c_2$.
$c_1$ and $c_2$ are the inverse radii of curvature, and
$|c_1| \le |c_2|$,
i.e. $c_1$ is the most nearly flat principle curvature.
A positive curvature indicates that the surface bends toward the
(outward pointing) normal vector $\vec N$ at the hit point.
A {\bf curvature} structure has three elements,
the unit vector {\bf crv_pdir} (or $\vec A$)
pointing in the direction of the first principle curvature,
the scalor {\bf crv_c1} (or $c_1$)
giving the curvature in the first principle direction, and
the scalor {\bf crv_c2} (or $c_2$)
giving the curvature in the second principle direction $\vec B$.
The second principle direction $\vec B$ is implied and can be found by
taking the cross product of the normal with {\bf crv_pdir}, i.e.,
$\vec B = \vec N \times \vec A$.

Each primitive solid can be considered to be bounded by
one or more {\em regular surfaces}.
Each regular surface is defined
as the locus of points $\vec S ( u, v )$
depending on two real parameters $u$ and $v$
which range from 0.0 to 1.0 inclusive.
These parameters form the coordinates of a two-dimensional Cartesian
$u,v$-plane.
A given $(u,v)$ coordinate
will appear only once on each regular surface,
but in objects with more than one surface
that same $(u,v)$ pair may appear at more than one place on the object.
The $(u,v)$ coordinate of the hit point is returned in {\bf uvcoord}
structure elements {\bf uv_u} and {\bf uv_v}.
For any hit point,
after the value of {\bf hit_point} has been computed,
the $(u,v)$ coordinates of that point can be
acquired by executing the C macro:
\beginverbatim
RT_HIT_UVCOORD( ap, stp, hitp, uvp );
\endverbatim

For some simple lighting-model applications, it is sometimes
desirable to create a mapping between the coordinate system on
the surface of an object to coordinates on a plane.
This is generally used to drive simple, two dimensional
{\it texture mapping}
algorithms.
The most common application is to extract a paint'' color
from a rectangular image file at coordinates $(u,v)$,
and apply this color to the surface of an object.
These parameters can also be used to simulate the effect of
minor surface roughness using the {\it bump mapping} technique.
Here, the $u$ and $v$ coordinates index into a rectangular file
of perturbation angles;
the surface normal returned by RT_HIT_NORM() is then
modified by up to $\pm 90$ degrees each in both the $u$ and $v$
directions, according to the stored perturbation.

In addition, the approximate beam coverage'' of the ray
in terms of the parameters $(u,v)$
is returned in the structure elements {\bf uv_du} and {\bf uv_dv}.
These approximate values are based upon the
ray's initial beam radius ({\bf a_rbeam}) and
beam divergence per millimeter ({\bf a_diverge})
as specified in the application structure.
These delta-$u$ and delta-$v$ values can be helpful for
anti-aliasing or filtering areas of the original texture map
to produce an area sample'' value for the hit point.

Combinatorial Solid Geometric (CSG) models are formed by the
boolean combination of primitive'' solids [MUUS87a].
For example, a plate with a hole is most easily modeled as
a plate primitive minus a cylinder primitive.
It is important to note that in CSG models, there is no explicit
representation of the surfaces of the solids stored;
indeed, for complex boolean combinations of complex primitives,
some of the resultant shapes may have very convoluted topology
and surfaces that may be at best high degree polynomials.

There are many applications that would benefit from being able
to express an {\it approximation} of
the complex shapes created using CSG modeling
as a collection of planar N-gons which together enclose roughly
the same volume of space as the original CSG solid.
The most obvious such application is to drive polygon-based
rendering routines (lighting modules) for predictive optical signatures.
On many modern workstations there is direct hardware or firmware support
for high-speed rendering of polygons [MOLN87].
In addition, there are whole collections of polygon-based predictive
The very best predictive radar signatures can be calculated using
the Method of Moments,
which requires having a three dimensional surface tessellation
to sub-wavelength resolution of the entire model.
A technique for computing this approximate three dimensional surface
tessellation is the focus of the majority of this paper.

such as the TRACK code of GTRI [PELF86],
do not operate
directly on a geometric representation of an object.
primarily due to the existence of dihedral and trihedral
structures in the object.
Rather than describing a vehicle simply as a collection
of these topological structures,
it is possible to analyze a three dimensional solid model
to locate all instances of
the topological features of interest.
For example, the software could locate
planar face elements; edges where two locally planar elements
join to make a dihedral, edges where three locally planar elements
join to make a trihedral, etc.
Then this list of topological features is used as input to the
feature-based analysis code.
% $Header: /vld/mike/talks/90nmg/RCS/joined.html,v 1.1 1998/05/01 06:53:13 mike Exp$

It should be abundantly clear that
it is highly desirable to have a single, central geometric database
which can be interrogated by a full suite of analysis codes.
Many fundamentally different kinds of model interrogation need to
be supported in order to meet this goal.
If a CAD system was being designed afresh, without {\it any} ties to
the past, then any underlying representation could be chosen and used.
In this section, the design considerations for expanding the kinds of
model interrogation available to the BRL-CAD Package
will be explored.

While the BRL-CAD Package comprises a large amount of
finished computer software,
the main investment is in the existing library of geometric models.
The amount of effort required to replicate these models
can be conservatively estimated in the hundreds of man-years.
In addition, the experience of the large number of
trained model designers and design analysts
is quite significant.
interrogation available within the system must protect these
investments.

The BRL-CAD Package as it stood in 1989 could be considered to be
one of the very few production CAD systems which was based
purely on the combinatorial solid geometry (CSG) technique.
The final shapes of all objects
were created by boolean combinations of primitive solids.
No attempt was made to represent in explicit form either the
topology or the surface geometry of any object.
The exact nature of the final form of an object was discovered
only on a point by point basis, by sampling the object with ray-tracing.
At the same time, the assortment of primitive solids available to
designers was definitely a hybrid combination of traditional
CSG primitives and more recent boundary-representation (B-rep)
primitives.
The primitive solids described by their boundaries included a
variety of faceted solids, as well as solids defined by
a closed collection of non-rational B-spline surfaces.
This rich collection of primitive solids could be combined by
the designer using any number of boolean operations.
However, complex combinations of primitives could be difficult
to visualize except through careful study of the three dimensional wireframe
approximation rotating in real time, coupled with the judicious use
of ray-traced renderings.

Given that modern computer workstations
[MOLN87]
are now commonly available with integral polygon rendering hardware,
and that hardware rendering speeds can exceed one million
polygons per second,
it seems highly desirable to be able to take advantage of this hardware.
Also, as discussed earlier, there is a burgeoning supply of
analysis software that simply can not make do with a
geometry interrogation interface that supports only the ray-tracing
What was missing from the BRL-CAD Package was a way of obtaining
an {\it explicit} description of the final shape of modeled objects.
Our quest then is for a technique suitable for obtaining
an explicit description of geometric objects.

There are several classes of modeling operations that are very
conveniently expressed in terms of boolean operations.
For example, holes can be bored in objects by subtracting a
drill bit'' solid from the original work piece,
and the primary shape of an injection mold can be created
by subtracting the desired final product from one or more
blocks of mold material.
Designers who have experienced the conceptual power of using boolean
operations to construct complex shapes are unlikely to want to
switch to a system that does not permit the use of boolean operations.
Given that the existing investment in geometric models depends
heavily on the use of boolean operations,
it was concluded that adding the capability for obtaining
an explicit description of modeled objects must provide
full support for the use of boolean operations.

This conclusion immediately places several rather severe requirements
on the design.
Most importantly, it requires that the underlying representation
used to hold the explicit description of the modeled objects
must be {\it closed} under boolean operations.
That is, given the explicit description of objects A and B,
then any boolean combination of A and B must be
representable as an explicit description expressed in
terms of the same underlying representation.
Several available choices of representation will be considered.

\eject
A strictly polygonal representation could be selected.
While performing boolean operations on solids described
as collections of polygons is not easy,
the representation can (with certain special definitions)
be considered to be closed under boolean operations, and
algorithms to accomplish the boolean evaluation
have been published for several years [LAID86].

A representation comprised exclusively of
rectangular parametric surfaces, such as B-splines
or similar tensor-product surface patches could be used.
However, research to date has shown that while B-spline
surfaces can be combined using boolean operations,
the resulting object can not be expressed strictly in
terms of B-splines [THOM84].
Instead, a mixed representation of B-splines and
polygons is produced, and this mixture becomes ungainly
when subjected to repeated boolean operations.
This occurs because the boolean combination of rectangular
parametric surfaces is not necessarily bounded by
rectangular parametric surfaces;  i.e., the representation is not closed.
Recent work has suggested that a representation comprised
of trimmed B-splines and shared-edge polylines might
be closed under boolean operations [COBB84],
but a full implementation is not yet known to exist.

No other good choices for an underlying representation
could be found.
Because the B-spline representation does not have closure
under the set of boolean operations, it regrettably could not be used.
Therefore, there was no choice;
the explicit representation of modeled objects would have
to be expressed in terms of collections of polygons.
This certainly met the requirements of the polygon based
application codes, but it posed a whole host of new questions.

Given that the explicit geometric representation of surfaces will be
approximated using polygons, there still remains the question of
how to deal with the representation of the topology.
The simplest strategy would be to simply ignore topology, and
store solid objects as collections of polygons.
This has several drawbacks because
these objects are intended to represent enclosed volumes of space.
It could be quite difficult to
verify that a given collection of polygons is in fact a valid solid,
without any cracks, dangling faces, or missing faces.
Also, implementing boolean operations without any explicit topology
would be nearly impossible.

Traditionally, solid modeling systems based on boundary representations
such as PADL [REQU82] have employed the {\it winged edge}
data structure for storing topology, as discussed earlier.
The disadvantage of the winged edge data structure is that
it is simply unable to represent many of the non-manifold conditions that
arise in the construction of complex shapes (such as finite element
meshes), and the non-manifold conditions that arise from the
use of the intersection operator ($\cap$) on two objects that
share only a single face, edge, or vertex.
Those systems employing the winged edge data structure that also
the use of {\it regularized boolean operators} which are defined
in such a way as not to produce any non-manifold results.
These regularized boolean operators are denoted by a superscript asterisk,
i.e., regularized union is $\cup ^ *$,
regularized intersection is $\cap ^ *$,
and regularized subtraction is ${-} ^ *$.

It seems unfortunate to have to restrict boolean operations on
the polygonal representation to not include any non-manifold results.
Some applications might desire them, and others might not;
unwanted non-manifold results are easy to discard,
but there is no easy way to re-infer the existance of
non-manifold results if the representation can not express them.
This is akin to the language limits thought''
concept from cognitive psychology.
Non-manifold results can be useful in a variety of ways.
One example is in interference or overlap checking.
The intersection of two objects can be computed.
If the intersection is the null set, then the two objects are
disjoint, and there is space between them.
If the intersection is a solid object, then the two objects
overlap in the given volume, which generally signals a modeling error.
If the intersection is a lone face, edge, or vertex, then
the two objects exactly touch, but do not overlap.

The inability of the winged-edge'' data structure to
represent non-3-manifold conditions prompted Weiler to
evaluate existing edge-based data structures [WEIL85].
This resulted in the development of a new data structure
suitable for representing
all the  Non-3-Manifold Geometric
(NMG)
and topological configurations
that boolean operations might produce [WEIL87].
This new data structure has been dubbed alternately
the radial-edge'', or NMG'' data structure.
Because of this structure's ability to handle
3-manifolds (solids), 2-manifolds (faces),
1-manifolds (edges), and 0-manifolds (points),
NMG objects are closed under boolean operations.

The NMG data structures have the advantages of complete generality
and closure under boolean operations,
plus they encode the full topological structure of an object,
as well as the geometric information.
The representation contains full topology information,
so that the relationships between vertices, edges, loops, faces, and
shells are continuously available.
Geometry is associated with each topological element.
For example,
the geometry information associated with a planar face is the plane
equation which includes the outward-pointing surface normal;
the plane equation does not have to be re-derived from the vertices.
This generality does come at a significant price in
increased memory use compared to the winged-edge data structure.
However, because of the anticipated frequency of occurance
of non-3-manifold conditions in CSG modeling, both intentionally
and as part of various analysis operations,
the NMG data structures were selected to contain
the approximate surface representation.

\eject
Several of the existing and planned primitive solids have
very complex curved shapes.
Some examples include the torus, the truncated generalized cone,
the hyperboloid of two sheets, the general polynomial solid,
and the B-spline solid.
In general, these solids have few or no sub-sections which are
flat.
Thus, these solids can not be represented by a collection of
polygons without loosing essential information about the
very nature of the solids.
Yet, the desire to have a homogeneous representation suggests
that all existing solids should be converted to a single new
underlying representation, so as to take advantage of the
boolean closure property, and to
enable applications to access the explicit surface representation.

The two options to consider are either (a) making a one-time
conversion of all existing models to a polygonal form,
and then performing all subsequent editing and processing
on a polygonal database; or (b) retaining the existing,
implicit combinatorial solid geometry database, and
providing some form of post-processing capability to
convert the implicit CSG shapes into an explicit, polygonal form.

Many existing application codes that use the ray-tracing paradigm
depend on being able to obtain very accurate surface normal
and Gaussian curvature information at any point on an object.
While surface normal interpolation [GOUR71]
can be used with a polygonal representation,
many artifacts can be introduced,
including surface normal discontinuities across polygon boundaries
and inconsistencies between the apparent and actual location
and shape of profile edges.
While these artifacts may be entirely tolerable in simple
rendering applications,
they are entirely inappropriate in a CAD system targeted for
high-resolution engineering analysis.
Finally,
it would be extremely distasteful to have to {\it reduce} the quality of
the ray-tracing style of interrogation in order to accomodate the
ability to obtain explicit object surfaces.

Therefore, the conclusion was that the existing combinatorial
solid geometry database would remain unchanged,
and that any application that required an explicit representation
of the modeled shapes would obtain that explicit representation
through some form of conversion of the underlying CSG database.
While this capability needs to be available to any application
at any time,
it can still be thought of as being a post-processing'' operation.

This choice of strategy has a number of fortunate implications.
Because a polygonal representation of a fundamentally curved shape
could never be more than an approximation of that shape,
it is impossible to choose a single resolution approximation
that would satisfy all applications.
Using a coarse approximation would quickly produce complaints
about a lack of accuracy in the modeling system.
For example, few engineering applications could tolerate octagonal
pipes being substituted for genuine round pipe.
On the other hand, using a very fine approximation would consume
gargantuan amounts of memory, which would impede simple operations
such as model editing and display.

With this strategy,
the polygonal representation can be considered just an {\it approximation}
of a much more accurate underlying geometric model.
Furthermore, because this approximation can be created
dynamically, each application has the opportunity to control the
resolution of the approximation being created.
This permits each application to obtain exactly the degree of
resolution required,
without having to worry about the approximation being too coarse
or too bulky.

The strategy that has been adopted has several key points.
Whenever an application requires an explicit description of
the surface of an object, it will access the {\it Application Interface}
to obtain an approximation created to meet accuracy requirements
of this particular application.
The explicit representation of the object will be returned
as a collection of planar faces embedded in the NMG data structures.

The existing Application Interface to the geometry database
already has a nice simple programming interface.
Thus, it seems appropriate to have a high-level interface
similar to {\bf rt_gettree()}.
The application would indicate what objects or object heirarchies
and what the accuracy requirements are,
and would be given in return a collection of NMG data structures
that would contain the surface approximation of the indicated objects.

\captionfig 4.5in by 3in, nmg-schematic, Figure 3 -- Schematic NMG Wiring Diagram.
%\figspace {3in} {4.5in} {X-3} {Schematic NMG Wiring Diagram.}

The actual implementation of
the application interface can be further broken into several pieces.
The existing routine {\bf db_walk_tree()} is used to retrieve
the indicated objects from the database.
needs to be converted to an approximate faceted form
stored in NMG data structures.
This operation is referred to as {\it tessellation}.
As each boolean operation is encountered in the database,
the appropriate tessellated solids will have that boolean
operation performed by the routine {\bf nmg_do_bool()}.
This routine will take the two tessellated objects
and combine them according to the boolean operation
back into a consistent set of solid tessellated objects.
Until very recently, it has been this step that has proven
the most difficult [LAID86].
combined with their boolean formulas, the resultant collection
of NMG objects is returned to the application.
(More precisely, the application is returned a {\bf struct model}
which contains pointers to the requested objects,
stored as one or more {\bf struct nmgregion}s).
This architecture gives rise to the schematic diagram
in Figure 3.

In the existing ray-tracing implementation,
{\bf librt} makes heavy use of an
object-oriented style of procedural interfaces to
geometry support routines.
defined a standardized set of operations that could
be performed on geometric objects.
The operations include
having a geometric object read itself
into memory, describe itself, produce a wireframe representation
of itself, and intersect a ray with itself.
This interface was extended to define a new operation
to require an object to tessellate itself into an NMG data structure.
This has the highly desirable property that
all the processing related to a given primitive solid
remains centralized in a single solid-specific geometry module.
a single module to the library;
none of the analysis codes ever need to be modified when
designers begin using a new kind of primitive.

From the schematic diagram it should be clear that the final evaluated
NMG solid object can be employed in a variety of ways.
The primary use will be for input to visualization and analysis software that
needs an approximate three dimensional surface mesh of the solid model.
However, a very powerful additional use will be to create new
faceted shapes which are then stored back in the database
as new geometric objects, suitable for future editing or analysis.

As will be explained in more detail in subsequent chapters,
each face of an NMG object will be composed of one or more planar N-gons,
each potentially non-convex and with embedded internal loops.
Applications that are prepared to deal with this topological
richness may operate on the NMG representation directly.
However, for those applications that require a simpler
face topology, a simplification routine exists that will
reduce each face to a collection of planar N-gons.
These simplified N-gons
may be non-convex but will have no embedded internal loops.
Finally, for applications that prefer faces to be collections
of simple triangles, a {\it triangulator} routine will be
provided that convertes the NMG faces into well-behaved
triangles [GOOD89].
One of the keys to the NMG approach lies with the concept of the radial-edge
representation as opposed to the classical winged-edge.  Within the
winged-edge representation, an edge represents the boundary between exactly
two faces.  The drawback to this is that an infinite number of planes or faces
can intersect in a single line in three dimensional space.  The winged edge
representation only allows for a pairwise topological connection between these
share the edge as a line of intersection.
The implementation described in this paper
is heavily patterned after a description of the radial-edge
data structures and operations written by Weiler [WEIL87].

\captionfig 3.0in by 2.5in, hierarchy, Figure 4 -- NMG Structure Hierarchy.

The basic topological
elements are the vertex, edge, loop, face, shell,
region, and model.
The relationship between these elements of the
hierarchy is depicted in Figure 4.
Note that for any element within the hierarchy, there is a direct path
from that element to the element which is one level higher,
and also to the element which is one level lower.

The {\it vertex} represents a unique topological point.
The {\it edge} is a line or curve in space terminated by
either one vertex, or two distinct vertices.
The {\it loop} is either a single vertex, or a circuit of one or more
edges.  A loop defines a circuit or a boundary of a space.
The {\it face} consists of one or more loops, and represents an actual
surface area.  The use of a loop within a face may define either an exterior
loop or an interior loop.  Exterior loops include an area in the face.
Interior loops exclude an area from the face surface, thereby causing a hole
in the face.

The {\it shell} is either a single vertex, or a collection of faces, loops,
and edges. The collection of faces in a shell may enclose a volume, thereby
creating closed objects, or may represent arbitrary surfaces.  Loops and edges
of a shell are refered to as wire loops'' and wire edges.''  They may be
used in creating wireframe aspects of the model.

The {\it region} is a collection of shells, and
the {\it model} is a collection of regions.

For the elements vertex, edge, loop, and face, there is a distinction between
the existence of the element and instances of the use of the element. This
allows multiple topological elements to share the same underlying form and
geometry.
The {\it vertexuse} is an instance or a use of a vertex.
The {\it edgeuse} is a directed instance of an edge.
The {\it loopuse} is an instance of a loop.
The {\it faceuse} is an instance or a use of a face.  Each side of
the face is uniquely represented by a faceuse, i.e.,
every face is referenced by exactly two faceuses.

Finally, note that each topological element makes reference to
a separate geometric element.
As a result of this separation of topology and geometry,
the kinds of geometric
support in the modeling system may evolve into richer and richer forms,
while continuing to enjoy a common set of topological elements
with a stable interface.
For example, the system described in this paper is
based upon manipulation of planar facets.
However, in the future
the geometry support of the system will be expanded to support curved
facets or B-splines, while retaining the same interface to the topology.

\vskip -22pt
%
The first {\bf long} (longword of memory) in any of the NMG
structures is
dedicated to a {\it magic number}.
It will be found either listed explicitly as the first
entry in the structure definition,
or it will be hidden,
obtained implicitly from the {\bf struct nmg_list}
sub-structure (described below)
which is the first entry in the structure definition.
Thus, it is always located at an offset of zero bytes from the
start of the structure.
This magic number serves a dual purpose.
First, every subroutine that is passed a pointer as a parameter
can dereference that pointer to obtain the magic number.
If the magic number obtained does not match the magic number
assigned for use with that kind of data structure,
then either memory has become corrupted,
or a defective pointer has been provided as a parameter.
Given that some NMG operations may have to dereference pointers
through seven connected data structures,
it is advantageous to detect invalid pointers as early
in the process as possible.
Second, some data structures employ {\it generic} pointers
which may refer to one of several different kinds of structures.
Rather than using an extra word of memory to store a type indicator,
the generic pointer can be dereferenced to obtain the magic number,
and thus the identity, of the referred-to structure.

A count of the number of structures within a model is kept.  Each structure
type contains an integer index'' member which uniquely identifies the instance
of a structure type within the model. This assists algorithms which must
temporarily associate some flag or bit of information with structures.  In
these instances, an array can be allocated with the appropriate size so that
there exists one array element for each structure in the model.  As the model
hierarchy is walked, the index elements can be used to quickly access the
temporary information appropriate for this particular structure.  Applications
for this include copying the model between memory and disk, and flags to help
assure that every structure is visited or operated upon exactly once.

The NMG data structures make frequent use of doubly linked lists.  With one
exception, they are all implemented using the same strategy and list
manipulation macros.
All linked lists are made up of nmg_list'' structures:

\vbox{ \beginverbatim
struct nmg_list  {
long            magic;  /* magic number */
struct nmg_list *forw;  /* forward'', next'' */
struct nmg_list *back;  /* back'', last'' */
};
\endverbatim }

The magic number field of this structure identifies the node as either a list
head or as a structural element.  The two pointers are to the
successor and predecessor of the node in the list.
Every list has one structure which is dedicated to functioning as
the head'' node.
An empty list consists of a head'' node
whose forw'' and back'' members are pointers to the head'' node.
Defining the doubly linked list as having an explicit head''
means that the enqueue and dequeue operations can operate on
any member of the list, and they do not need to refer to the head.

Consider the task of making a linked list of {\bf edgeuse} structures.
The first element of the {\bf edgeuse} structure is an
{\bf nmg_list} structure named {\bf l}.
Thus, the address of {\bf l} is a {\it pun} (or homonym) for
the address of the {\bf edgeuse} structure.
By adding {\bf l} to a linked list, in reality the whole {\bf edgeuse}
This allows the list manipulation functions
to be generalized to handle lists of any kind of structures.
The manipulation
routines merely operate on {\bf nmg_list} objects and need not know details
about what structure types are in the list.
The only requirement is that the {\bf nmg_list} structure must
appear as the first element in the containing structure.
A rich set of macros exist for insulating the programmer
from all the details of inserting and deleting elements from
a list, walking a linked list, and various initialization
and clean-up operations.

When creating variable names and structure member names, the
implementation makes heavy use of abbreviations.  As a result, it was
important to regularize the abbreviation strategy.  The suffix _p'' is
appended to the end of all pointer variables in the structures.  The first
characters of the variable indicate the type of object the pointer
references.  For example, a pointer to a vertex structure would be v_p'', and
vu_p'' would be a pointer to a vertexuse structure.

Some structures types may have a variety of different structure types as
parent'' or child'' structures.  Since each structure maintains a pointer to
its parent and children, a method for maintaining a syntactically correct
handle for such objects is required.  Such pointers are stored as unions of
pointers to each possible type of structure required, plus a pointer to a
magic number.  For example, Figure 4 demonstrates that a vertexuse may
have either a shell, a loopuse or an edgeuse for a parent.  As a result, the
vertexuse structure will contain something similar to the following:

\vbox{ \beginverbatim
union {
struct shell   *s_p;
struct loopuse *lu_p;
struct edgeuse *eu_p;
long           *magic_p;
} parent;
\endverbatim }

This union provides a handle for each possible type of parent for the
vertexuse structure.  In addition, it contains a pointer to magic number''
handle.  This allows the type and validity of the parent to be identified
as a particular structure type,
{\em before} the parent structure is referenced.
This union owes
its existence to the Pascal origins of the implementation by Weiler.
In the C language, a simple pointer to long integer'' and the language's
ability to {\it coerce} one type of pointer into another type of pointer
using type-casting would have been sufficient.  It is debatable whether
type-casting or the union-name-handle approach

The names for linked list head nodes'' use the _hd'' suffix, and the first
letters indicate the type of object in the list.  For example,
the head of a list of vertexuse
structures would be called vu_hd''.
When a structure is to be a part of a linked list, the first
element of the structure will be
an element {\tt l}'', the list node which becomes a member of a linked list.
As described in
the section on linked list implementation, by keeping this item as the first
element in the list, the list manipulation interface can be made fully general
for lists of all types of elements.

% was 3.75 by 3.5
\captionfig 2.5in by 2.35in, vertex, Figure 5 -- Vertex and Vertex Uses.

The simplest element in the system is the {\it Vertex}.
A vertex represents a
single point within the topological space of the object being modeled.  It
also serves as a linkage point for connecting the topological model with the
geometrical data.
The structures vertex'' and vertexuse'' can be conceptually viewed as in
Figure 5.
The magic number is stored directly as the first element of the structure.
The second item is an nmg_list'' substructure.  This substructure forms the
head of a doubly-linked list of all the {\it uses} of this vertex. The member
vg_p'' is a pointer to the geometry.  The index element keeps the structure
index for the particular vertex instance.
The vertex structure is referenced through a vertexuse structure.

\setbox0=\vbox{ \beginverbatim
struct vertex {
long            magic;
struct nmg_list vu_hd;
struct vertex_g *vg_p;
long            index;
};
struct vertexuse {
struct nmg_list     l;
union {
struct shell    *s_p;
struct loopuse  *lu_p;
struct edgeuse  *eu_p;
long            *magic_p;
} up;
struct vertex       *v_p;
struct vertexuse_a  *vua_p;
long                index;
};
typedef double point_t[3];
struct vertex_g {
long     magic;
point_t  coord;
long     index;
};
\endverbatim }

The l'' element of the vertexuse structure
is entered on the vertex structure's vu_hd'' list, which is a list of all
uses of the vertex.
This linked list node also contains the magic number for the vertexuse
structure.
A vertexuse may be needed by any of the higher level objects: shell, loopuse,
or edgeuse.  The union up'' contains a pointer to each of these three
structure types.  It also contains a pointer to a magic number, which can
be used as a handle for getting the magic number of the parent.
The element v_p is a pointer to the vertex being {\it used} by this
vertexuse.
The vertexuse may have an associated vertexuse attribute structure.
This structure is referenced through the vua_p'' pointer.
One example of such information might
be to store a surface normal for each vertexuse,
suitable for intensity interpolation shading algorithms
or for normal-vector interpolation shading algorithms.

The vertex geometry structure is very simple and straightforward.  It consists
of a magic number, the coordinates of the vertex, and a structure index.
The coordinates are stored in a variable of type point_t, which is an array
of three double-precision floating point numbers.

% Was 5in by 2.25in;  page is only 4.5in wide!
\captionfig 4in by 1.75in, edge, Figure 6 -- Edge and Edgeuse.

The next topological element is the {\it edge}.  The edge represents a line
or curve between a pair of vertices.  The unique members of this structure
are eu_p'' which is a pointer to one of the uses of the edge, and eg_p''
which is a pointer to the edge geometry.
The edge geometry structure is reserved for
future curved-edge support.
Note that the edge structure itself does not reference the endpoints of the
edge.  The endpoints are accessed through the edgeuse structure
because almost all references to the edge endpoints occur while
processing the edgeuse structures [WEIL87]. Within the edgeuse structure, the
edge struct pointer e_p'' ties the edgeuse to the appropriate edge.

\setbox0=\vbox{ \beginverbatim
struct edge {
long            magic;
struct edgeuse  *eu_p;
struct edge_g   *eg_p;
long            index;
};
struct edgeuse {
struct nmg_list     l;
union {
struct loopuse  *lu_p;
struct shell    *s_p;
long            *magic_p;
} up;
struct edgeuse      *eumate_p;
struct edge         *e_p;
struct vertexuse    *vu_p;
long                index;
};
\endverbatim }

% was 4.75in by 3.9in, page is only 4.5!

Each edgeuse references one vertex via the vertexuse structure pointer
vu_p''.  The other end of the edge/edgeuse is referenced through the first
edgeuse's {\it mate} edgeuse.  The pointer eumate_p'' connects an edgeuse to
it's edgeuse mate.  The eumate_p edgeuse is conceptually the use of the edge
on the opposite side (interior/exterior wise) of the face from the existing
edgeuse. The eumate_p'' and radial_p'' pointers form a linked list of the
radial uses'' of an edge. The edgeuse referenced by eumate_p'' is on the
opposite loopuse/faceuse of the same loop/face.
The edgeuse l'' list node is used to
form loops of edges,
or to keep lists of all of the wire edges'' which are a
part of a shell.
Like the vertexuse, the edgeuse can be referenced by one of two different
types of structures higher in the hierarchy.  The same technique is used for
providing handles for these parent'' structures
as was used for the vertexuse.

% was 4 by 5.
\captionfig 2.75in by 3.5in, looptypes, Figure 8 -- Variations of the Loop.

A loop defines a boundary or circuit.  Conceptually, a loop consists of either
a single vertex, or a series of one or more edges in a circuit.  Like the
edge, most of this information is stored in the use structure.
The loop structure member lu_g'' is a pointer to the geometry support for
the loop.
The member lu_p'' connects the loop structure to one of the
loopuses of the loop.  From that loopuse, the other use of the loop is reached
through the lumate_p'' structure pointer.
The geometry structure continas a bounding box for the loop.

\setbox0=\vbox{ \beginverbatim
struct loop {
long            magic;
struct loopuse  *lu_p;  /* Ptr to one use of this loop */
struct loop_g   *lg_p;  /* Geometry */
long            index;
};
struct loopuse {
struct nmg_list     l;
union {
struct faceuse  *fu_p;
struct shell    *s_p;
long            *magic_p;
} up;
struct loopuse      *lumate_p;
char                orientation;
struct loop         *l_p;
struct nmg_list     down_hd;
long                index;
};
\endverbatim }

The loopuse structure is where most of the loop details are handled.  The
first element is a linked list node, so that faces and shells may keep track
of collections of loopuses.  The union up'' provides a
handle for all possible parents of the loopuse.
The lumate_p'' member provides a pointer to the other use of the same loop as
this loopuse.  Often the other loopuse will be the use of this loop on the
opposite surface of a face.  The orientation'' member is used to define
whether this loopuse is being used to enclose space, or exclude space within a
face. The loopuse is associated with a particular loop through the pointer
lu_p.''
The structure member down_hd'' is a list head which is used to access the
component elements which form the loop.  In the case where the loop is made up
of one or more edges, this list will consist of a series of edgeuses.  When
the loop is formed on a single vertex,
the first and only item in the list will be the
vertexuse associated with the loopuse.

\captionfig 3in by 3in, loopedges, Figure 9 -- Orientation of Edgeuses within a Loopuse.

Figure 8A
depicts a use of a loop (a loopuse) where the boundary formed by the loop
consists of a single vertex.
Figure 8B
depicts a loop which
is formed of one edge, and
Figure 8C
depicts a loop which
is formed of four edges.
When the loopuse is made up of edgeuses, they will be arranged to form a
circuit.  The mates to these edges will form a circuit for the loopuse mate.
If the loopuses are a part of a faceuse, the edgeuses are arranged in a
special orientation.  When the cross product of the vector of the edgeuse and
the normal vector for the faceuse is taken, the resultant vector should point
into and along the surface of the face.  See Figure 9.

%\clearpage
The face represents a planar or curvilinear two-dimensional boundary or
surface.
The pointer fu_p'' is a pointer to one of the two faceuses of the face,
the other being reached through that faceuse's fumate_p'' pointer.  The
pointer fg_p'' references the geometry structure for the face.
The face geometry structure keeps a bounding box, and
the plane equation of the face.
The equation
is stored as a 4-tuple of double precision floating point numbers.
A point $\vec A$ which lies on the plane will satisfy the following
relation:
$$N_x * A_x + N_y * A_y + N_z * A_z - N_3 = 0$$

\vbox{ \beginverbatim
struct face {
long                magic;
struct faceuse      *fu_p;
struct face_g       *fg_p;
long                index;
};
\endverbatim }
\vbox{ \beginverbatim
typedef double plane_t[4];
struct face_g {
long                magic;
plane_t             N;
point_t             min_pt;
point_t             max_pt;
long                index;
};
\endverbatim }

Faceuses represent a side of the surface of a face.  Each face will therefore
have exactly two faceuses associated with it.
The first element in the faceuse structure is a linked list node.  This is
here so that the shell may keep a list of all the faceuses which are in the
shell.  The pointer s_p'' is a pointer to the parent shell of the faceuse.
The other use of the same face is indicated by the faceuse pointer
fumate_p''.  The orientation'' member indicates which side of the face is
represented by the faceuse.  It actually indicates whether the surface normal
of the faceuse is the same as the normal stored in the face geometry
structure, or whether the normal is reversed.
The faceuse is associated with a particular face through the f_p'' pointer.
lu_hd'' keeps track of all loops in the face.

\vbox{ \beginverbatim
struct faceuse {
struct nmg_list     l;
struct shell        *s_p;
struct faceuse      *fumate_p;
char                orientation;
struct face         *f_p;
struct nmg_list     lu_hd;
long                index;
};
\endverbatim }

The shell represents a set of collected, inter-related (and probably
connected) items.  The first element is a linked list node to allow groups of
shells to be collected into a list.  The member r_p'' is a pointer to the
parent nmgregion for the shell.  The attribute or geometry structure for the
shell as a whole are refered to through the pointer sa_p''.
The lists fu_hd'', lu_hd'', eu_hd'' are used to keep track of the faceuses,
wire loopuses, and wire edgeuses which make up the shell.  The pointer vu_p''
points to a single vertexuse when the shell consits of a single vertex.
The shell attribute structure sa_p'' is used to store a bounding box
for the entire shell.

\vbox{ \beginverbatim
struct shell {
struct nmg_list     l;
struct nmgregion    *r_p;
struct shell_a      *sa_p;
struct nmg_list     fu_hd;
struct nmg_list     lu_hd;
struct nmg_list     eu_hd;
struct vertexuse    *vu_p;
long                index;
};
\endverbatim }

\eject
Regions are used to keep collections of associated shells within the model
space.
The nmgregion is included in the model's list of regions through the
Because the structure name region'' was already in extensive use
through out the BRL-CAD Package when the development of the NMG capability was
begun, the structure has been called nmgregion'' instead of simply
region''.
At some point in the future, the implementation will
probably be altered so that all NMG structures begin with the prefix nmg''.
The nmgregion'' structure consists of a linked list node l'',
a pointer to the parent model m_p''
and a list of shells which make up the region s_hd''.
The region attributes structure
is used to store a bounding box for the entire region.

\vbox{ \beginverbatim
struct nmgregion {
struct nmg_list     l;
struct model        *m_p;
struct nmgregion_a  *ra_p;
struct nmg_list     s_hd;
long                index;
};
\endverbatim }

The model represents the top of the hierarchy for the NMG structures.  The
list r_hd'' keeps track of all regions or nmgregions'' in the model space.
The long integer maxindex'' contains the number of structures which have been
allocated in the model.  It exists so that arrays can be allocated with one
element for each structure in the model.

\vbox{ \beginverbatim
struct model {
long                magic;
struct nmg_list     r_hd;
long                index;
long                maxindex;
};
\endverbatim }

There are several differences between this implementation of the radial
edge data structures and Weilers implementation [WEIL87].
Some of the differences are a result of the
different programming languages used, while others are a result of the
different goals of the implementations.
While Weiler was implementing an
entire modeling system based exclusively upon the NMG structures,
within the BRL-CAD Package NMG objects are just one representation
of many.
The differences are outlined as follows:

\beginitemlist

\listitem Wire loops are permitted as members of the shell to
simplify the creation and manipulation of loops and faces within the system.
This also permits a shell to be comprised of a point cloud'' made up of loops
on individual vertex points.

\listitem Each of the structures faceuse, loopuse, and edgeuse as described by
Weiler had attribute substructures which were used for storing information
unique to a particular use of the object.  This has not proven necessary in
this implementation.

\listitem The shell keeps a list of all wire edges within the shell.  Under the
system described by Weiler, some wire edges were discovered'' by looking at
vertices (vertexes) used in the shell, to find other uses of the same
vertex(es) which were children of edgeuses whose parent is
the same shell.

\listitem The linked lists in the Pascal language structures described by Weiler
were maintained by directly manipulating next'' and last'' pointers within the
structures.
In this implementation,
all structures which are kept in lists contain
a generic list node'';
manipluation of the linked lists is via a standardized set of macros.

\listitem In this implementation, generic pointers which may reference more than one
type of structure are disambiguated by
dereferencing the pointer and inspecting resulting magic number''.
Because Pascal is a strongly typed'' programming language,
Weiler's implementation required that a structure pointer be disambiguated
before it was dereferenced.
Hence an additional variable in the structure containing the
pointer was used to indicate the appropriate interpretation for the generic
pointer.

\listitem All uses of a face must belong to one shell.

\enditemlist
The applications programmer is insulated from the details of the NMG
radial-edge data structures by a library of functions which perform all of the
basic tasks.  Each of the routines accepts a valid NMG model or a part of one,
and performs an operation, returning a valid model upon completion.
The routines within the library can be classified as constructive,
destructive, or manipulative.  This chapter provides a basic description of
the routines in these categories.
Within the library, all routines which are intended for use by the
applications programmer or user begin with the nmg_'' prefix.

The basic constructive routines all have the letter m'' (which stands for
make'') immediately after the nmg_'' prefix.

\beginitemlist
\listitem struct model *nmg_mm()
{\it Make Model''} creates a new NMG model structure and fills in the
appropriate fields.  The result is an empty model.

\listitem struct model *nmg_mmr() {\it Make Model, Region''} creates a new
model with a call to nmg_mm'' and creates a region within the model.  The
region is empty.

\listitem struct shell *nmg_msv(struct nmgregion *r_p) {\it Make Shell,
Vertex''} creates a new shell consisting of a single vertex in the parameter
region.  A new vertex is created for the shell.

\listitem struct nmgregion *nmg_mrsv(struct model *m) {\it Make Region, Shell,
Vertex''} creates a new region within the existing model, and creates a shell
of a single vertex within the region.

\listitem struct vertexuse *nmg_mvvu(long *upptr) {\it Make Vertex,
Vertexuse''} exists so that shells, loops and edges can be created on a new
vertex.  upptr'' points to the parent
structure for which the vertex and vertex use are being created.  The new
vertexuse will reference this structure as its parent.

\listitem struct vertexuse *nmg_mvu(struct vertex *v, long *upptr) {\it Make
Vertexuse''}
allocates a new vertexuse for an existing vertex.  The vertexuse becomes a
child of the structure indicated by upptr''.

\listitem struct edgeuse *nmg_me(struct vertex *v1, struct vertex *v2, struct
shell *s) {\it Make Edge''} creates a new wire edge in the shell specified.
The vertex pointer parameters may either indicate vertexes for the endpoints
of the new wire edge, or may be NULL pointers.  New vertex structures will be
generated for each NULL vertex parameter.

\listitem struct edgeuse *nmg_meonvu(struct vertexuse *vu) {\it Make Edge on
existing Vertexuse''} is used to create an edge on a vertex in a loop or
shell.  The resultant edge has the same vertex at each endpoint.

\listitem struct edgeuse *nmg_eusplit(struct vertex *v, struct edgeuse *eu)
\break
{\it Edgeuse Split''} splits an existing edgeuse pair
of a wire or dangling'' face-edge by inserting a new vertex.

\listitem struct edge *nmg_esplit(struct vertex *v, struct edge *e) {\it Edge
Split''} causes a new vertex to be inserted along an existing edge.  If the
parameter vertex pointer is NULL, an new vertex is created.  The vertex
inserted need not lie along the existing edge.  See Figure 10.

\listitem struct edgeuse *nmg_eins(struct edgeuse *eu) {\it Edge Insert''}
inserts a new, zero length edge between the edge associated with the parameter
edgeuse and the edge associated with the edgeuse previous'' to the parameter
edgeuse. See Figure 10.

\captionfig 4.75in by 1.8in, edgefunc, Figure 10 -- Edge Operators.

\listitem struct loopuse *nmg_ml(struct shell *s) {\it Make Loop''} takes the
largest possible number of contiguous wire edges which form a circuit from the
parameter shell and uses them to create a wire loop in the shell.

\listitem struct loopuse *nmg_mlv(long *magic, struct vertex *v, int orientation)
{\it Make Loop, Vertex''} creates a new vertex-loop.  The loop will be a
child of the structure indicated by the magic number pointer parameter, and
will have the specified orientation.  If the vertex pointer is NULL, a new
vertex is created for the loop.

\listitem struct faceuse *nmg_mf(struct loopuse *lu1) {\it Make Face''}
generates a new face from the parameter wire loop and its mate.

\enditemlist

In order to simplify the creation of manifold and non-manifold faces, The
following two routines are provided for the application developer.

\beginitemlist
\listitem struct faceuse *nmg_cface(struct shell *s, struct
vertex **vt, int n) {\it Create Face''} creates a (non-manifold) face from a
list of vertex structure pointers.  The face will be a child of the shell
indicated in the parameter list.  The parameter vt'' is an array of pointers
to vertex structures.  The length of the array is indicated by the parameter
n''.  If vt'' is a null pointer, then the face will be created as a
polygon on n'' new vertex structures.  If vt'' is non-null, a null entry
in the list will cause a new vertex to be allocated for that position.  The
vertexes in the list should be ordered in a clockwise manner for a viewer
looking backwards along the normal vector of the desired face.

\listitem struct faceuse *nmg_cmface(struct shell *s, struct vertex **vt[~], int
n) {\it Create Manifold Face''} generates a manifold face in the indicated
shell.  The parameter vt'' is an array of n'' pointers to pointers to
vertex structures. If a pointer in vt is a pointer to a null vertex pointer, a
new vertex is created for that position.  In this way, new vertices can be
created conveniently within a user's list of known vertices. The vertices
should be listed in clockwise'' order for a viewer looking backwards along
the normal of the desired face.  In addition to creating the face, the routine
will join edges of the new face with dangling edges of other faces in the same
shell. This makes it easier for the applications code to generate
topologically correct, closed, manifold objects. \enditemlist

\eject
The usual technique for creating a model is to first allocate a model with a
call to the Make Model'' routine nmg_mm''.  The resultant model does not
have any regions within it. Next, a region, and shell consisting of a single
vertex are created with a call to Make Region, Shell, Vertex'' or
nmg_mrsv''.
This lone vertex will be consumed when the first edge or loop is
created in the shell.

At this point, there are many separate paths to creating the first face.
If a manifold object is the desired goal, then one of the routines
nmg_cface'' or nmg_cmface'' should probably be employed.
Alternatively, a loop consisting of a single vertex can be created with a call
to nmg_mlv'' (Make Loop, Vertex).  Then a face can be made with the loop via
a call to nmg_mf''.  The loop may then be expanded to an edge-loop with calls
to nmg_meonvu'' and new vertex points inserted with nmg_esplit''.

% was 4.5 by 2
\captionfig 3.5in by 1.75in, cmface1, Figure 11 -- Conceptual View of Arguments to nmg_cmface.

Ordinarily, the applications programmer will be interested in creating
simple closed objects.  The final topology of the object is already well
understood at the time of creation.  That is to say, the number of vertices
is known, as well as their relationships to various edges and faces.  In
order to simplify the creation of such objects, the library provides the
routine {\it nmg_cmface} or {\it "Create Manifold Face."}  In addition to
providing a simple interface for creating complex faces, the routine also
takes care of meshing the edgeuses of the new face with the radial edge
structure of pre-existing edges.  This is important for the creation of
topologically closed objects.

To build a face using {\it nmg_cmface}, the application programmer must
provide a pointer to the parent shell for the face, an array of pointers to
pointers to struct vertex, and the length of the array.  The reason for the
complexity of the array is actually to simplify the job of the applications
programmer when making faces for which vertices do not yet exist.

The application keeps an array of pointers to struct vertex (the array {\it
verts} in the example below). This array contains pointers to the vertex
structures used in the object being built.  A null pointer in this list
indicates a vertex structure which does not exist prior to the creation
of the face or object.  When these pointers are encountered by the
nmg_cmface routine, a new vertex will automatically be allocated and a
pointer to the new vertex inserted in the list.  The mechanism for this will
become clear later.
Another array ({\it vertp} in the example below) is used as the argument to
{\it nmg_cmface}.  This second array will contain pointers to elements in
the first array.
For example, take as given an array of
pointers to struct vertex called {\it verts[~]}, and an array of pointers to
pointers to struct vertex called {\it vertp[~]}.
Now suppose a triangular face is desired, and verts[0], verts[2], and
verts[3] are pointers to the vertices at the corners of the face.
The result can be conceptually viewed as in Figure 11. The
elements of {\it vertp} should be ordered so that when the new face is
viewed from the outside or normal-ward direction, the vertices will be
listed in clockwise order.  The face can then be created in the shell
indicated by the pointer {\it shell_p} with the following call to
nmg_cmface:

\vbox{ \beginverbatim
struct vertex  *verts[4], **vertp[3];
vertp[0] = verts[0];
vertp[1] = verts[2];
vertp[2] = verts[3];
nmg_cmface(shell_p, vertp, 3);
\endverbatim }

Thus to create an NMG object (as is done in the tesselation process),
the application first builds the list of vertex structures
needed.  This consists of pointers to pre-existing vertex structures, and
null pointers for new vertex structures.  Then each face is constructed in
turn.

When the model or some part of the model is no longer needed, the structures
involved must be deleted.  Each of the functions described below kills'' or
deletes a structure type, and all its children.
Normally, the only destruction routine called directly by the application is
nmg_km'', which is called when the model is no longer needed.

\beginitemlist

\listitem void nmg_km(struct model *m) {\it Kill Model''} deletes an entire NMG
model.

\listitem void nmg_kr(struct nmgregion *r) {\it Kill Region''} deletes an
nmgregion and all of its children.

\listitem void nmg_ks(struct shell *s) {\it Kill Shell''} removes a shell and
all of its children.

\listitem void nmg_kfu(struct faceuse *fu1) {\it Kill Faceuse''} deletes a
faceuse, its mate and the face which they share, as well as all components
which made up the faceuses and face.

\listitem void nmg_klu(struct loopuse *lu1) {\it Kill Loopuse''} removes a
loopuse, it's mate, and all children.

\listitem void nmg_keu(struct edgeuse *eu) {\it Kill Edgeuse''} removes an
edgeuse, and its mate from a radial edge, and deletes them.  If these were the
last radial uses of the edge, then the edge structure and children are deleted
as well.

\listitem void nmg_kvu(struct vertexuse *vu) {\it Kill Vertexuse''} removes a
vertexuse from the list of uses of the vertex and deletes the vertexuse
structure.  If this was the last use of that vertex, the vertex structure and
children are deleted as well. \enditemlist

These routines mold and manipulate the NMG structures.

\beginitemlist

\listitem void nmg_face_g(struct faceuse *fu, plane_t p) {\it Face Geometry
construction''} assigns a plane equation to the face of the given faceuse.

\listitem void nmg_face_bb(struct face *f) {\it Construct Bounding Box for
Face''} by looking at (and computing) the bounding boxes for all the
constituent loops.

\listitem void nmg_vertex_gv(struct vertex *v, pointp_t pt) {\it Assign
vertex Geometry''} to a vertex.

\listitem void nmg_loop_g(struct loop *l) {\it Compute Geometry} (bounding
box) {\it for loop''}.

\listitem void nmg_shell_a(struct shell *s) {\it Compute Shell Attributes''}
(bounding box) for a shell.

\listitem void nmg_region_a(struct nmgregion *r) {\it Compute Region
Attributes''} computes the bounding box for the nmgregion from all constituent
shells.

\listitem void nmg_movevu(struct vertexuse *vu, struct vertex *v) {\it Move
Vertexuse''} causes a vertexuse to be moved to a new vertex.

\listitem void nmg_unglueedge(struct edgeuse *eu) {\it Un-Glue Edge''} removes
an edgeuse and its mate from a shared radial edge, and creates a new edge for
them to share.

\listitem void nmg_moveeu(struct edgeuse *eudst, struct edgeuse *eusrc) {\it
Move Edgeuse''} causes an edgeuse eusrc'' and its mate to become uses of a
new edge.  In the new edge, eusrc'' will be immediately radial to eudst''
in the radial edge list of that edge.  If eusrc'' and its mate were the last
uses of the old edge, then the old edge structure and it's children are
deleted.

\listitem void nmg_moveltof(struct faceuse *fu, struct shell *s) {\it Move
first wire Loop in shell to an existing Face''}.  Not normally called by
applications programmer.  The first wire loop (loopuse pair) of the specified
shell is moved to the given faceuse (and faceuse mate).

\listitem void nmg_jv(struct vertex *v1, struct vertex *v2) {\it JoinVertexes''} causes all uses of vertex v2'' to become uses of v1''.  Vertex
v2'' is deleted.

\listitem void nmg_isect_faces(struct faceuse *fu1, struct faceuse *fu2) {\it
Intersect Faces''} causes two faces to be intersected and the line of
intersection to be identified and inserted.

\listitem void nmg_mesh_faces(struct faceuse *fu1, struct faceuse *fu2)
{\it Mesh Faces''} performs the topological and geometrical meshing of two faces
which share one or more edges of intersection.  When complete, opportunities
for edge sharing between the two faces will be taken advantage of, and radial
edge orientations of the two faces will be appropriate for all shared edges.

\listitem struct model *nmg_find_model(long *magic_p) {\it Find Model''}
returns the parent model for a given structure.  The structure is identified
by a pointer to its magic number''.

\listitem struct vertexuse *nmg_find_vu_in_face(point_t pt,
struct faceuse *fu,
\break
fastf_t tol) searches for a vertex with geometry coordinates within
tolerance in a particular face.

\listitem struct nmgregion *nmg_do_bool(struct nmgregion *s1, struct nmgregion
*s2, int oper, fastf_t tol) performs a specified boolean operation using the
indicated regions as inputs.  Produces a region as output.

\listitem int nmg_ck_closed_surf(struct shell *s) {\it Check for Closed
Surface''} tries to determine if the parameter shell represents a closed
object.

\listitem int nmg_manifold_face(struct faceuse *fu)
returns a non-zero value if
the parameter face represents a part of a closed surface.

\listitem int nmg_demote_eu(struct edgeuse *eu)
turns an edgeuse into a pair of
loopuses.  Each loopuse consists of a single vertexuse.  The edgeuse and its
mate are deleted, and if they were the last use of their edge, the edge and
its children are delete.

\listitem int nmg_demote_lu(struct loopuse *lu)
turns a loop into a collection
of wire edges. The loopuses of this loop are deleted, as is the loop and its
children.

\listitem void nmg_simplify_loop(struct loopuse *lu) \\
void nmg_simplify_face(struct faceuse *fu) \\
void nmg_simplify_shell(struct shell *s)
all serve to simplify the topology of the faces in
a shell. Unnecessary edges within a face are deleted, and the adjacent loops
are joined together.  This service is primarily required to simplify the
topology generated as a result of boolean operations.

\listitem void nmg_jl(struct loopuse *lu, struct edgeuse *eu) {\it Join
Loops''} combines loops which share a common edge.  Useful primarily as a
support routine for the Simplify'' routines.

\listitem char *nmg_identify_magic(long magic)
returns a string that describes the name of the structure
type associated with the given magic number.
This is most useful in program diagnostics associated with
error detection and recovery.

\enditemlist
% $Header: /vld/mike/talks/90nmg/RCS/joined.html,v 1.1 1998/05/01 06:53:13 mike Exp$

The job of a tessellator is to convert a given solid primitive into
a faceted approximation stored in NMG data structures.
There are two aspects to this conversion:  establishing the
topology of the approximation, and then generating the geometry to
associate with the approximate topology.

Conversion of a faceted primitive solid will be exact, with each face,
edge, and vertex of the NMG representation one-to-one
with a topological element in the
original representation.
In general, effecting the conversion of a faceted primitive to the
NMG representation is direct.
Difficulties usually arise only when
the original representation did not contain enough topology information
to permit direct generation of the NMG topology.
For example, the {\bf PolySolid} bag of polygons'' solid contains a
collection of faces which together are known to enclose one or more volumes,
yet there is absolutely no explicit topology.
For this solid, the tessellator routines must rediscover the topology
of the original object.  This is done by a geometric comparison of
each vertex against all the vertices previously encountered in this solid.
If the vertex has been previously seen,
then a new {\em use} of the existing vertex is made.
If the vertex is unique,
then a null pointer is added to the parameter list for {\bf nmg_cmface()} so
a new topology vertex can be created.
When all the vertices of a face have been processed in this manner,
the list of vertices are passed to the routine {\bf nmg_cmface()},
which connects all the vertices together with edges.
If an edge existed between two vertices in the new face, then
a new use of the existing edge is created, otherwise an entirely new edge
is created.
Finally, these edges are formed into a loop, and the loop is embedded
in a new face.

Conversion of curved, implicitly defined primitive solids to
a faceted representation will necessarily be inexact.
To provide control over the nature and magnitude of the errors
that may be introduced by the faceted approximation used
in the tessellation, three types of tolerances are passed to the
tessellator.
The {\em absolute}
tolerance, which limits the maximum permissible difference between
any point on the tessellation and the corresponding point on the
original solid, is expressed as an absolute distance.
In the subroutine interface, this absolute distance is passed as
a double precision floating point value, given in millimeters.
In the {\bf mged} user interface, this absolute distance is specified
in terms of the current working units (millimeters, inches, etc).
The absolute tolerance permits users to make absolute statements about
the maximum distance error contained in any tessellation.
For example, using this mechanism it is possible to ensure that
no face deviates from the true surface by more than 2 mm.
The {\em relative} tolerance also limits the maximum error of any point,
but is expressed as a fraction between 0.0 and 1.0 of the
diameter of the bounding sphere which encloses the original solid.
This relative tolerance permits users to make statements about
the relative error contained in any tessellation.
For example, this could ensure that
no face deviates from the true surface by more than five percent of the size
of the solid.
The {\em normal} tolerance limits the maximum angular error of the surface
normal.  This normal tolerance permits users to make statements about
the accuracy of the surface normals.
For example, this could ensure that the surface normal of all faces
do not deviate from the exact surface normals of the
corresponding points on the original solid by more than 5 degrees.

The tolerances can be set singly or in any combination.
If no tolerances are set, each tessellator module establishes a default
minimum tessellation;  for example, most tessellators will not approximate
a circle with fewer than 6 line segments.
If more than one tolerance is specified, then on a solid by solid basis,
the most restrictive tolerance is applied.
If both an absolute and a relative tolerance are given,
then large solids would most likely be tessellated to satisfy the absolute
tolerance, while small solids would most likely be tessellated to
satisfy the relative tolerance.
For example, if an abs=10mm and rel=1\%,
a large sphere of diameter 1000mm would be subjected to the absolute tolerance,
while a small sphere of diameter 10mm would be subjected to the relative
tolerance.

\captionfig 3in by 3in, torus, Figure 12 -- Calculating Arc/Chord Error.
The topology of the torus is a rectangular mesh where the edges''
of the mesh are connected together, side to side, and top to bottom.
This can be visualized as follows:
if the torus is cut once and straightened, it becomes a right circular
cylinder.
If the cylinder is cut longitudinaly and uncurled, it becomes a rectangle.
The length'' of this rectangle is the distance around the rim of the torus
and is traversed as the angle $\alpha$ varies from $0$ to $2\pi$.
The width'' is the distance around one cross-section of the torus
and is traversed as the angle $\beta$ varies from $0$ to $2\pi$.
The torus can also be constructed by constructing a circle in $\alpha$
with radius $r_2$,
and then sweeping that circle out around $\beta$ with radius $r_1$.

The torus tessellator must determine the minimum number of facets that
can be used to represent the torus while still satisfying the given
error tolerances.  Fortunately, the two parts of the torus are separable,
and both parts are circles, so this problem reduces to determining
the fewest number of line segments that can be used to approximate a
circle while still meeting the error tolerances.

Consider a triangle
inscribed inside a circle of radius $r$.
One vertex of the triangle is at the
center of the circle $O$;
the other two touch the circle at points $A$ and $B$,
as shown in Figure 12.
The line segment $AB$ subtends an angle
$$\angle AOB = \theta$$
The point of maximum error is $C$, the midpoint of $AB$:
$$C = {{A + B} \over 2}$$
$$\angle AOC = {\theta \over 2} = \gamma$$
Extending the line $OC$ until it hits the circle results in the point $D$.

Satisfying the surface normal tolerance is the easiest.
If the circle is divided into line segments, the surface normal
is exact at the line segment midpoint $C$, and has the largest error
at the endpoints of the line segment: $A$ and $B$.
The surface normal error at the endpoints is $\theta \over 2$.
The relationship between the number of line segments $n$ and the
angle $\theta$ that each line segment subtends is
$$n = {2\pi \over \theta}$$
Thus, to satisfy a surface normal error tolerance of $ntol$,
the error at the endpoints
$${\theta \over 2} \leq ntol$$
Therefore the minimum number of line segments that can be used is
$$nseg = { {2\pi} \over {2 ntol} } = { \pi \over ntol }$$
The face distance error inherent in the linear approximation of the circle is
$$e = | D - C | = r (1 - cos { \theta \over 2 } )$$
Thus, to meet the face distance error tolerance,
a choice of $\theta$ must be made so that
$e$ satisfies the relation
$$e \leq dtol$$
The maximum value of $\theta$ which can be used is
$$\theta = 2 cos^{-1} ( 1 - { dtol \over r} )$$
and the minimum number of line segments that must be used is:
$$nseg = {{2 \pi} \over \theta} = { {\pi} \over { cos^{-1} ( 1 - { dtol \over r} ) } }$$

To efficiently satisfy the maximum surface error tolerance,
it is necessary to find the minimum number of line segments
that must be used in each of the length'' and width''
directions.
The appropriate radius is substituted into the formula for $nseg$
to determine
$nlen$ (the number of segments needed in the length direction), and
to determine $nw$ (the number of segments needed in the width direction).
The largest of either of $nlen$ or $nw$ or
the number of segments required to satisfy the surface normal
tolerance is used.

All the vertices on the surface of the torus can be generated by
varying $len$ from 0 to $nlen$ while also varying $w$ from 0 to $nw$,
and computing:
$$\alpha = w { { 2 \pi } \over nw }$$
$$\beta = len { { 2 \pi } \over nlen }$$
$$R = A cos(\beta) + B sin(\beta)$$
$$P = V + R + r_2 {R \over |R|} cos(\alpha) + H sin(\alpha)$$

where
\bigskip
\centerline{\vbox{
\halign{\hfil#\ & #\hfil\cr
V	& vertex point at the center of the torus \cr
A, B	& perp. vectors, lie in the plane of the torus, define length'' \cr
G, H	& perp. vectors, define width'' direction \cr
H	& normal to plane of the torus \cr
P	& surface point \cr
$r_2$	& radius of torus around the rim \cr
}}}

\setbox0=\vbox{
{\tt \beginverbatim
E
/\
/  \
G/____\H
/\    /\
/  \  /  \
/____\/____\
D      I     F

\endverbatim }
\centerline{Figure 13 -- Initial Octahedron Face}
}

An ellipsoid is defined by a vertex point $V$ at the center
and three mutually perpendicular vectors $A$, $B$, and $C$
which define the eccentricities.
Through an affine transformation, the ellipsoid can be
mapped to a unit sphere located at the origin.
The algorithm begins by approximating the sphere
by an octahedron, with the six vertices at
$V \pm A$, $V \pm B$, and $V \pm C$.
Each face of the octahedron is a triangle $\triangle DEF$,
as in Figure 13.

Points DEF line on the surface of the unit sphere.
Pick the points GHI as the midpoints of the three edges of ABC.
If each of these points GHI are viewed as vectors from the origin
in the direction of the surface of the unit sphere,
then re-normalizing the vectors to have unit length will cause the points
to lie on the surface of the unit sphere.
Consider each of the four new triangles
$\triangle DGI$, $\triangle GHI$, $\triangle IHF$, and $\triangle GEH$
recursively until the tolerance is satisfied.

\setbox0=\vbox{
{\tt \beginverbatim

E
/\
/  \
J/____\K
/\    /\
/  \  /  \
G/____\/____\H
/\     L    /\
/  \        /  \
/    \      /    \
/      \    /      \
/        \  /        \
/__________\/__________\
D            I           F

\endverbatim }
\centerline{Figure 14 -- Partially Subdivided Octahedron Face}
}

It is tempting to consider an adaptive subdivision algorithm,
so that when the magnitudes of A, B, and C are not equal,
the areas of higher curvature could be tessellated more finely.
Unfortunately, it is not possible to use different levels of
subdivision without introducing cracks'' into the tessellation.
Consider the case where triangle $\triangle GEH$ needs further subdivision,
but triangles $\triangle DGI$, $\triangle GHI$, and $\triangle IHF$ do not,
as in Figure 14.
The problem here arises because the edge $GH$ in $\triangle GHI$
will no longer match up with edge $GL$ in $\triangle GJL$
and edge $LH$ in $\triangle LKH$, because point $L$ will have
been normalized out to meet the unit sphere.
While cracks could be prevented with by splitting $\triangle GHI$ into
$\triangle GLI$ and $\triangle LHI$, this would produce an
irregularity in the topology of the tessellation that would make
a non-recursive formulation significantly more difficult.

Let
$$r = max( |A|, |B|, |C| )$$
Then, the distance tolerance $dtol$ can be expressed as a
maximum angular (normal) tolerance as before:
$$\theta = 2 cos^{-1} ( 1 - { dtol \over r} )$$
The final tolerance to use is the more strict of the
surface normal tolerance $ntol$ and the distance tolerance:
$$tol = min( ntol, \theta )$$
Thus, the number of triangles to be used around any one
circumference of the ellipsoid will be
$$nseg = { \pi \over ntol }$$
However, because the initial approximation to the ellipsoid is
an octahedron, the number of segments must be a multiple of four.

The coordinates of all the vertices of the tessellation of a unit
sphere with $nseg$ triangles are computed.
Each of these coordinates is then transformed back into the
coordinate system of the ellipsoid.

To evaluate a boolean combination of two tessellated solids,
three distinct sub-operations must be performed.
First, all elements of the two shells must be intersected.
Second, every element in the two shells must be classified
as {\bf in}, {\bf on}, or {\bf out}.
Finally, all undesired elements are eliminated.

\eject
The first step in the boolean operation process is the intersection and
cutting of the two shells with respect to each other.  This process is
summarized in Figure 15.

\setbox0=\vbox{ {\tt\beginverbatim
if bounding boxes of shell A and shell B overlap
for each face in shell A,
if bounding box of face A overlaps bound box of shell B
for each face in shell B
if bounding boxes of face A and face B overlap
intersect edges of face A with plane of face B
intersect edges of face B with plane of face A
insert new topology & perform meshing
for each wire edge in shell A,
if wire edge A overlaps bounding box of shell B
for each face in shell B
if edge A intersects face B
if necessary split edge at plane,
insert vertex at plane intersection into face
for each wire edge in shell B
if wire edges intersect
create any needed verticies in both wire edges
\endverbatim}
\centerline{Figure 15 -- Shell Intersection Procedure.}
}

The process of comparing the face bounding boxes involves
a check to see
if the bounding boxes overlap along the line of intersection, not just
whether they overlap at all.  This is important for reducing the number of
face intersections performed.

When the bounding boxes of two faces overlap, they must be intersected with
each other.  The plane equations of the two faces are compared for equality.
If the two faces are coplanar, they must have their loops intersected using a
two dimensional polygon clipping approach.
If the two faces are not coplanar, they must be intersected so that the faces
will share an edge at the intersection.
A list of all vertexuses or points which are on the line of intersection
between the two planes is first generated.  To do this, each edge of face A is
intersected with the plane of face B and each edge of face B is intersected
with the plane of face A.

This intersection process takes the following form:  If an endpoint of the
edge is either topologically or geometrically in the plane of the other face,
that vertex is registered in the list of points on the line of intersection.
If the span of the edge crosses the plane of the other face, then the edge is
divided into two edges.  The new vertex which divides the edge lies on the
line of intersection and is added to the list of points along the line of
intersection.

\captionfig 3in by 3in, seginsert1, Figure 16 -- Segment Insertion: Both Points in Face.

After each face has been intersected with the plane of the other face, the
resulting list of points of intersection is sorted geometrically along the
line of intersection.  The list is then used to determine which segments of
the line of intersection are shared between both faces.  Such segments must be
added to each face.  For each face, a segment to be added will fall into one
of three categories:
(1) both endpoints are in the face,
(2) one endpoint is in the face, or
(3) no endpoints are in the face.
In the first case, a check is made to make certain that the edge does not
already exist.  If it does not, then each of the two endpoints are examined.
If they are part of the same loop of the face, then that loop is divided into
two separate loops, which share a common (new) edge.  If the vertices belong
to different loops, then an edge which connects the two loops is created, and
the two loops are joined into a single loop.
Examples of each of these cases can be seen in Figure 16.

%Was 4.75 by 2.25
\captionfig 3in by 1.5in, seginsert2, Figure 17 -- Segment Insertion: One Point in Face.

% Was 4.75 by 2.0
\captionfig 3in by 1.2in, seginsert3, Figure 18 -- Segment Insertion: No Points in Face.

When only one of the endpoints exists in the face, the loop which contains the
existing vertex is extended to include the new point.  This can be seen in
Figure 17.
Finally, when neither of the segment endpoints exists in the face, a new
interior loop must be created in the face as in Figure 18.

When all of the appropriate topology has been inserted into the faces, the
topology of the two faces is connected so that edges and vertices which can
be shared between the faces are indeed shared.  This process consists mainly
of combining vertexuses onto a common vertex, and arranging the radial-edge
orientation of edgeuses about a shared edge.

Wire edges of the shell must also be intersected with the other shell.  Wire
edges exist either as part of a wire loop, or as an individual wire edge.
Both types are processed in the same manner.  Each edge is compared to the
bounding box of the other shell.  Where there is overlap, the edge is
intersected with each face and wire of the other shell.  If the edge
intersects any of these, the edge is split if necessary, and the point of
intersection is topologically linked to the other face or wire.

When all of the intersections have been performed, every object in each
shell must be classified with respect to the other shell.  Each face, loop,
edge, and vertex must be classified as being inside, on the surface of
(referred to here as on''), or outside of the other shell.  Here the topology
becomes helpful.  This process is easier if each shell is classified in turn
against the other shell.  For notational clarity, we refer to the
classification of an object in shell A with respect to shell B.

The classification of all structures is stored in a single table.  The index''
fields in the NMG structures serve as offsets into the table.  Once the
classification is complete, an individual structure's classification can be
quickly gotten by using the index'' field from the structure as the offset
into the table of classifications.

If by, referring to the topology, it can be determined that the vertex has
previously been classified against shell B, then obviously any vertexuse of
that vertex which is in shell A shares the same classification.  If it has not
previously been classified, it is classified now.

By looking at the other vertexuses of a vertex, it is possible to determine
if there is a use of the vertex in the topology of shell B.  If such a use
exists, then the vertex, and the current vertexuse are classed as being on''
shell B.

If the topology does not indicate that the vertex is on'' shell B, the
geometry must be used to determine if the vertex is inside'' or outside'' of
shell B.  First of all it should be obvious that all vertices which lie
outside of the bounding box of shell B are easily classified as being outside
of shell B.  Should this fail to classify the vertex, a raytracing approach is
employed.

A single ray is fired'' from the vertex along an arbitrary line
and is intersected with all of the faces in shell B.
Typically, this line will be along one of the major axis
directions, although it may be useful to send the ray in some other,
non-aligned direction to reduce the probability of hitting the edge of a face
or hitting the plane of a face edge-on.
The number of {\em Manifold} faces which the ray encounters
before leaving the bounding box of shell B is counted.  If the number of
crossings is odd, then the vertex is classified as inside'' of shell B.
While this appears simple at first glance, there are several problems involved
in the raytracing approach which are worth noting:
(1) identifying a face as either manifold or non-manifold,
(2) recognizing an actual hit on a face, and
(3) coping with a ray hit on an edge or a vertex.

If a face has any dangling edges, it can be quickly classed as being
non-manifold.  This is a relatively quick check.
If for each edge of the face, there exists an odd number of other face-loops
of manifold faces of the same shell adjacent to the same edge, then the face
is manifold.  This is a tedious check which requires an exhaustive analysis of
most, if not all of the faces of the shell to determine if one face is
manifold.  While this is important for some cases, most of the time it is not
necessary to perform a full manifold check of the radial faces.  A check for
dangling edges of radial faces is sufficient.

\captionfig 3in by 3in, rayisects, Figure 19 -- Raytracing Hit-Points.

Since faces consist of both interior loops (holes in the face) and exterior
loops, it is important to be sure that a ray which hits the plane of a face
actually hits inside the surface area of the face as well. The in/out status
for the hit-point is determined by the loop which has the sub-element (edge or
vertex) closest to the hit-point.

If the hit-point is closest to an exterior loop of the face,
then (a) the ray
hits the face if the point is inside the loop,
as in hit-point 1 in Figure 19A,
and (b) the ray misses the face if the point is outside the loop.
Likewise,
if the hit-point is closest to an interior loop,
then (c) the ray misses the face if
the hit-point is inside the loop, as in hit-point 3 in Figure 19A,
and (d) the ray hits the face if the hit-point is outside the loop,
as in hit-point 2 in Figure 19A.

\sectionheadfour{Intersecting a Ray and Edge Or Vertex}
When the hit-point of the ray lies on an edge or vertex of a face, further
logic must be used to determine whether or not a hit actually occurs.  If the
ray hits an edge of a loop in a face, and there is another loop of the face of
the same type (interior/exterior) adjacent on the edge, the hit is registered
as if the ray had intersected the interior of the loop.  For example, hit-point
4 in Figure 19A is not a hit on the face because the edge is shared
between two interior loops of the same face.  On the other hand hit-point 5 in
Figure 19A is a hit on the face because the edge being hit is shared
between two exterior loops of the face.

If the edge is actually a boundary of the face, then the ray must be compared
to the surface normals of the faces involved.  For example, the ray V1 in
Figure 19B scores a single hit as it encounters the edge between
the two faces.  Ray V2 does not score a hit.  It is important to note that
only one hit may occur for the pair of faces, since it was the edge between
them which was encountered.
The situation becomes slightly more complex when there are more than two faces
of the shell sharing the edge.  In this case, the hit/miss status for each
pair of faces is determined at the time the edge is first encountered, and all
faces are marked as having been processed.

Once all the vertex structures in shell A have been classified against shell
B, it is possible to classify the edges.  First cut classification of edges is
done by referring to the classification of the endpoints.  If one or more of
the endpoints is not on'' shell B, the edge takes it's classification from
that vertex.  If both endpoints are classified as on'' shell B, the mid-point
of the line segment is computed, and classified using the raytracing technique
described in the previous section.  An edge with endpoints inside and outside
shell B indicates an error in the intersection process.
Table 1 summarizes the classification of edges.

\vbox{
\vskip 10pt
\centerline {
\vbox{ \offinterlineskip
\hrule
\halign{\vrule\strut\ \hfil#\hfil\ \vrule&
\ \hfil#\hfil\ \vrule\cr
Endpoint&	Edge\cr
classifications&classification \cr \noalign{\hrule}
both out&	out \cr \noalign{\hrule}
out/on&		out \cr \noalign{\hrule}
both on&	use mid-point ray \cr \noalign{\hrule}
in/on&		in \cr \noalign{\hrule}
both in&	in \cr \noalign{\hrule}
in/out&		intersection error \cr
}
\hrule
} % vbox end
} % centerline end
\vskip 22pt
\centerline {Table 1 -- Edge Classification by Endpoint Classification}
}

A loop of a single vertex inherits the vertex's classification.  A loop of
edges which contains an edge which is not on'' shell B inherits that
classification.  For example, if a loop contains an edge classified inside''
shell B, then the loop is classified inside'' shell B.  A loop with edges
both inside and outside shell B indicates an error in the intersection
process.
If all edges of a wire loop are classified as on'' shell B,
the loop is classified as on'' shell B.
There are other conditions required for a face loop to be classified as being
on'' shell B.  First, there must exist a loop in the topology of shell B which
has exactly the same set of edges.  Secondly, the loop must be classified as
being either shared'' or anti-shared''.  A shared'' face loop not only has a
counterpart in the other shell, but the normals of the faces of which the
loops are a part of are pointing in the same direction.  A loop is
anti-shared'' when the surface normals of the parent faces point in opposite
directions.

Faces are not classified except to the extent to which the loops which make up
the face are classified.
% $Header: /vld/mike/talks/90nmg/RCS/joined.html,v 1.1 1998/05/01 06:53:13 mike Exp$

\captionfig 3.75in by 1.75in, op-intro, Figure 20 -- Classification of Example Objects.

After all the topological elements in objects A and B have been
classified, the boolean evaluation simply becomes a task of
deciding which elements to retain, and which to destroy.
Every element has been classified with respect to both objects A and B,
and was assigned one of eight combined classifications.
The object from which the element originally came from is always
given an {\bf on} classification.
Elements from A are classified as one of
{\bf onAinB, onAonBshared, onAonBanti-shared, onAoutB,}
while elements from B are classified as one of
{\bf inAonB, onAonBshared, onAonBanti-shared,} or {\bf outAonB}.
All of these possibilities occur in the two example object pairs
(four blocks viewed end-on)
in Figure 20; the important face-loop classifications are
appropriately labeled.

For the two blocks on the left of Figure 20,
the bottom center face-loop
exists in both objects A and B, and
the orientation (outward surface normal) of the face-loop in both objects
is the same (pointing towards the bottom of the page),
so the face-loop is classified {\bf onAonBshared}.
For the two blocks on the right of Figure 20,
the middle face-loop
also exists in both objects A and B,
but the orientation of the two face-loops are opposite,
so this face-loop is classified {\bf onAonBanti-shared}.

\vbox{
\vskip 10pt
\centerline {
\vbox{ \offinterlineskip
\hrule
\halign{& \vrule#&
\strut\ #\ \hfil\vrule&
\ #\ \hfil&
\vrule#\ \vrule&
\ #\ \hfil\vrule&
\ #\ \hfil\vrule&
\ #\ \hfil&
\vrule#\cr
&A&   B&&		A -- B&	     A $\cup$ B& A $\cap$ B& \cr \noalign{\hrule}
&on&  in&&		kill&	     kill&	 retain&	\cr
&on&  on shared&&	kill&	     retain&	 retain&	\cr
&on&  on anti-shared&&	retain&	     kill&	 retain&	\cr
&on&  out&&		retain&	     retain&	 kill&	\cr \noalign{\hrule}
&in&  on&&		retain+flip& kill&	 retain&	\cr
&on shared&	on&&	kill&	     kill&	 kill&	\cr
&on anti-shared& on&&	kill&	     kill&	 kill&	\cr
&out&  on&&		kill&	     retain&	 kill&	\cr
}
\hrule \vskip 3pt } %vbox end
} %centerline end
\centerline{Table 2 -- Boolean Evaluation Decision Table}
\vskip 10pt
}

% was 3.75 by 1.5
\captionfig 3.5in by 1.5in, op-sub, Figure 21 -- Subtraction Performed on Example Objects.

Because there are only eight possible classifications,
the appropriate action for the boolean evaluation algorithm
can be easily tabulated.
These actions are found in Table 2.
Indeed, this exact same table exists in the source code
for the boolean evaluator in file {\bf nmg_eval.c}.
Placing these actions into a table permits a complete separation
of {\em policy} and {\em mechanism} in the implementation.
The policy is encoded in the entries of the table,
and the mechanism is embodied in the code of the subroutine.
This has two main benefits.
First, code to perform any one action exists in only one place,
ensuring consistent treatment of all cases.
Second, this offers the potential for adding more
boolean operations at a later date, simply by supplementing the table
with another column.

Consider first the case of the subtraction operation, A minus B.
The intent is to retain every part of A that is {\bf in} only A,
to kill every part of A that is also {\bf in} B,
and to kill every part of B.
The policy just stated is implemented by the tabulated rules for
processing all the topological elements.
The entry for an element which has classification {\bf on} A
and {\bf out} B has a table entry of retain''
(denoted more suscinctly as {\bf onAoutB}=retain).
This rule preserves the main body of A.
{\bf onAinB}=kill and {\bf onAonBshared}=kill
because these elements are {\bf in} B,
and therefore represent a portion to be subtracted out.
{\bf outAonB}=kill to eliminate unused parts of B.
{\bf onAonBanti-shared} elements are retained,
because they are on the surface of A and on the surface of B,
where the two surfaces touch.
To implement subtraction, this case has to be defined as either
(a) shaving an infinitesimally thin layer off of the surface of A,
or (b) a no-op, where the A surface is retained unmodified, and
the B surface is killed.
Because choice (a) would modify the volume of the resulting solid,
it can not be used;  instead, choice (b) has been selected.
This policy is implemented in these three table entries:
{\bf onAonBanti-shared}=retain,
{\bf onBonAshared}=kill, and
{\bf onBonAanti-shared}=kill.
Finally, elements which have been classified as
{\bf inAonB} are listed in the table as retain+flip''.
The surface has to be retained because it will become part of the
new boundary between A and the outside world.
However, the existing surface normal points {\em into} solid A,
reflecting the fact that this surface was originally part of the
exterior of solid B, so the surface normal must be flipped.
The results of performing a subtraction on the two sample objects
is illustrated in Figure~21.

While the discussion of the table entries has been made in terms
of surfaces defined by face-loops,
the same reasoning and actions apply to the inferior topological
elements:
wire-loops, loop-edges, wire-edges, edge-vertices, and lone-vertices.
The overall strategy is to process all the topological elements
of object A, and then to process all the elements of object B.
The meat of the algorithm exists in internal subroutine
{\bf nmg_eval_shell()},
which starts by processing the loops in each face.
Any loopuse which has a classification that maps to
an action of kill'' is demoted into
a collection of wire edges using {\bf nmg_demote_lu()}.
The loop is demoted rather than killed so that
edges and vertices that are shared in common with both objects
can be properly considered later in the subroutine.
If none of the loops in the face are retained, then the faceuse
is killed using {\bf nmg_kfu()}.
If some loops in the face are retained, then the faceuse is retained;
if the faceuse came from object B then it is moved to object A,
the faceuse's mate is also moved to object A,
and the face normal is flipped.
The algorithm then proceeds down from faces to processing wire-loops.
If the loopuse has a classification that maps to
an action of kill'' then it is demoted into
a collection of wire edges,
otherwise it is retained, and moved into object A if necessary.
Next, wire-edges are processed.
If the edge is marked kill'' it is demoted into vertices
using {\bf nmg_demote_eu()},
otherwise it is retained, and moved into object A if necessary.
Finally, lone-vertices are processed.
Vertices marked kill'' are disposed of using {\bf nmg_kvu()}.
Because vertices are 0-manifolds there is no lower topological
element to demote them to.

\captionfig 3.75in by 1.5in, op-union, Figure 22 -- Union Performed on Example Objects.

Consider next the case of the union operation, A $\cup$ B.
The intent here is to retain all elements that are on the
exterior of either A or B, while eliminating any interior structure
or redundant elements.
More precisely, for solid modeling, the formula for union is interpreted as
$$A \cup B ~ := ~ (A - B) + (B - A)$$
where the $+$ operation is a simple combination, or sum, operation.
The elements that are on the exterior of exactly one of either A or B
are classified as {\bf onAoutB} and {\bf outAonB} and are retained.
Elements that are on the exterior of both A and B appear twice,
first as {\bf onAonBshared} which is retained,
and again as {\bf onBonAshared}, which is killed (to avoid having
the element become duplicated in the result).
Interior structure is found in all elements classified as
{\bf onAinB}, {\bf inAonB}, plus any anti-shared faces
{\bf onAonBanti-shared} and {\bf onBonAanti-shared}.
All elements with these classifications are killed.
The result of performing the union operation on the two example
objects is shown in Figure 22.

\captionfig 2.5in by 0.75in, op-inter, Figure 23 -- Intersection Performed on Example Objects.

Finally, consider the case of the intersection operation, A $\cap$ B,
as shown in Figure 23.
The intersection operation retains all elements that are
simultaneously part of both A and B, while discarding the excess.
Clearly, elements classified {\bf onAonBshared} and {\bf onAonBanti-shared}
are elements common to the two objects and should be retained;
elements classified {\bf onBonAshared} and {\bf onBonAanti-shared}
need to be killed to prevent duplication.
The elements exterior to one of the objects are
classified {\bf onAoutB} and {\bf outAonB} and should also be killed.
Previously interior structure {\bf onAinB} and {\bf inAonB}
should also be retained, as these elements will form the new
boundary when the non-common elements have been removed.

The technique just described for evaluating a boolean formula
is simple to program and debug, works reliably,
and is easy to understand.
Storing all the policy decisions in a table
makes the algorithm very straightforward.
% $Header: /vld/mike/talks/90nmg/RCS/joined.html,v 1.1 1998/05/01 06:53:13 mike Exp$

The majority of a BRL-CAD user's contact with the new non-manifold
geometry capability will be through the Multi-Device Graphics
Editor ({\bf mged}) [APPL88].
This is the program that is used
to rapidly view existing geometry in wireframe form,
to create new geometry and update existing shapes,
to select viewpoints for ray-tracing runs,
to select keyframes for animation sequences,
and to preview animation sequences in wireframe form.

At any time in an {\bf mged} session, the user may add one or more
objects to the active model space, using the command
\beginprose
mged> {\em e object}
\endprose

If the viewing cube is
suitably positioned, the newly added subtrees become visible on the display.
The normal mode of operation is for users to work with wireframe
displays of the unevaluated primitive solids.  These wireframes can be
created from the database very rapidly.

On demand, the user can request the calculation of
approximate boundary wireframes that account for
all of the boolean operations specified along the arcs of the
directed acyclic graph in the database.
The evaluation of the approximation is performed by
tessellating each of the primitive solids into an NMG object
meeting the current tolerance, and combining them according to the
indicated boolean operations.
Each edge and vertex in the resulting NMG object are placed
in a {\bf struct vlist} chain (by subroutine {\bf nmg_r_to_vlist()})
and drawn on the {\bf mged} display.
This operation is invoked with the command
\beginprose
mged> {\em E object}
\endprose
or
\beginprose
mged> {\em ev -w object}
\endprose

Evaluating the surface takes somewhat longer than
creating wireframes of the unevaluated primitive solids,
so it is not used by default for viewing large assemblies.
However, it is quite reasonable to evaluate the surface whenever the
design has reached a new plateau, or
when a detailed examination of a complex part is required.
When viewing objects of modest complexity at loose tolerances,
the evaluation process is quite rapid.

Note that the evaluated boundary wireframes are not stored in the database,
and are primarily intended as a visualization aid for the designer.

% Figure \ref{crod} shows an engine connecting rod.
% On the left side is the wireframe of the unevaluated primitives
% that the part is modeled with, and on the right side is the approximate
% boundary wireframe that results from evaluating the boolean expressions.

On those hardware platforms where polygon drawing capabilities exist,
it is possible to have a flat-shaded polygonal rendering of
database objects drawn.
This operation is invoked with the {\em evaluate} command
\beginprose
mged> {\em ev object}
\endprose

Once the polygonal rendering of the object is on the screen, it
can be rotated in real time.
This capability gives the designer the opportunity to more fully
appreciate the complex shape which has been created,
and to judge whether the evaluated shape matches the intended design.

On hardware platforms which have hardware support for
rendering lit and shaded polygons with multiple light sources
(such as the Silicon Graphics 4D workstations),
it is possible to activate the hardware lighting model.
This provides a much more realistic rendition of the evaluated objects,
and also gives clues about face normals.

When further used in conjunction with hardware clipping planes,
the evaluated solids can provide significant new insight
for the designer.

As an aid to the user, an option to the {\em ev} command exists
to add surface normal vectors to the evaluated surface polygon display.
The outward pointing surface normals are drawn as single vectors
from the centroid of each polygon.
They resemble whiskers'' on the polygons.
This option is invoked with the command
\beginprose
mged> {\em ev -n object}
\endprose

If the polygon rendering of the object shows all the faces,
but no surface normal vectors can be seen, then this suggests
that either an error exists in the database entry for this object,
or an internal software error in {\bf librt} has been encountered.
In either of these cases, the vectors should be pointing inside
the solid object.
This can be easily verified by using the hardware clipping planes
to permit the inside of the solid to be seen.
If the vectors are on the inside, then the surface normals
have become reversed.

As discussed the in section on tessellation,
there are three types of tolerances that are user selectable.
The absolute tolerance ensures that no point on the polygonal
approximation will be further away from the true surface than
the tolerance value.
The relative tolerance ensures that no point on the polygonal
approximation will have a distance from the true surface which
is greater than the given fraction of the primitive object's size.
The surface normal tolerance ensures that no face normal at any point
on the polygonal approximation will differ from the true surface normal
at the corresponding point on the true surface by more than the
indicated angular error.
If multiple tolerances are given, the tesselator is required to
satisfy all of the tolerances.
This enables the user of the approximation to make firm statements
about the upper bound on the tessellation error produced.

When {\bf mged} is first started, a relative tolerance of one percent
is the default tolerance.
At any time the tolerance settings can be examined, using the {\em tol}
command.  Thus, immediately after starting {\bf mged}, the {\em tol}
command would show:
\beginprose
mged> {\em tol}
Current tolerance settings are:
abs None
rel 0.01 (1\%)
norm None
\endprose

The absolute tolerance specification is given in the current working
units, and is converted internally into millimeters.
For example, adding an absolute tolerance of three millimeters might
be done like this:
\beginprose
mged> {\em tol abs 3}
mged> {\em tol}
Current tolerance settings are:
abs 3 MILLIMETERS
rel 0.01 (1\%)
norm None
mged> {\em units inches}
mged> {\em tol}
Current tolerance settings are:
abs 0.11811 INCHES
rel 0.01 (1\%)
norm None
\endprose

The absolute tolerance can be disabled by setting the tolerance to zero
(or a negative number).
Providing a tessellation with zero error is impossible;
in general, to create a tessellation of a curved object with
zero error would require an infinite number of planar faces.
Therefore, zero tolerance can be used as a flag to turn off the
absolute tolerance requirement.

The relative tolerance specification is given as a fraction between zero
and one.
\beginprose
mged> {\em tol rel .001}
mged> {\em tol}
Current tolerance settings are:
abs 3 MILLIMETERS
rel 0.001 (0.1\%)
norm None
mged> {\em tol rel 0}
mged> {\em tol}
Current tolerance settings are:
abs 3 MILLIMETERS
rel None
norm None
\endprose

The surface normal tolerance is given as a positive angle in degrees between
zero and 90.
Specifying a normal tolerance of zero is used to disable the normal
tolerance requirement.
The angular tolerance is echoed back both in fractional degrees,
and as degrees minutes and seconds of arc, as can be seen in the
second example below.
\beginprose
mged> {\em tol norm 2}
mged> {\em tol}
Current tolerance settings are:
abs 3 MILLIMETERS
rel 0.01 (1\%)
norm 2 degrees (2 deg 0 min 0 sec)
mged> {\em tol norm 0.5}
mged> {\em tol}
Current tolerance settings are:
abs 3 MILLIMETERS
rel 0.01 (1\%)
norm 0.5 degrees (0 deg 30 min 0 sec)
\endprose

If a model included parts that had been created from the boolean
combination of both very large and very small primitives,
where the small primitives have a size within a few orders of
magnitude of the desired absolute tolerance,
it may be desirable to specify both an absolute and a relative
tolerance.
For the large objects, the absolute tolerance would ensure that
the tessellation produced enough facets to represent the surface
to the desired degree of accuracy, whereas the relative tolerance
would probably have generated significantly fewer facets.
For the small objects, the absolute tolerance would probably not
generate very many facets, but the presence of
a relative tolerance would ensure that the tessellation produced
a reasonable rendition of the desired shape.

The surface normal tolerance will also limit the error
in the tessellation.
However, the number of faces in the tessellation can grow rapidly
as the tolerance is tightened.
For example, consider tessellating a single torus.
If no normal tolerance is given, the torus is approximated by
36 ($6 \times 6$) facets, and a normal tolerance of
one degree yields 32400 facets.
Other values are listed in Table 3.
While specifying a tight tolerance on the surface normal will
produce tessellations with a pleasingly smooth surface,
the large number of facets that result can consume substantial
amount of memory.
If all tolerances are disabled (including the default one percent
relative tolerance), then each tessellator will choose an
object-specific minimum number of facets needed to produce
a reconizable rendition of the primitive solid.

\vbox{
\vskip 5pt
\centerline {
\vbox{ \offinterlineskip
\hrule
\halign{&\vrule#&\strut\hfil#\hfil& \hfil#&\vrule#\cr
&Tolerance &Facets	&\cr\noalign{\hrule}
& none	& 36	&\cr
& 10	& 326	&\cr
& 5	& 1296	&\cr
& 2	& 8100	&\cr
& 1	& 32400	&\cr
& 0.5	& 129600	&\cr}
\hrule \vskip 5pt
} %vbox end
} % centerline end

\centerline{Table 3 -- Relationship Between Normal Tolerance and Facets for Torus}
}

During the development of the boolean evaluation software,
several graphical displays were created to assist in
the debugging process.
By default, this feature is not active, but setting a run-time
debugging flag allows any user to enable it.

The first display shows the process of intersecting two faces together.
The outline of all the loops in the first face are drawn.
Performing the intersection operation can result in
Adding new edges may also result in the cutting or joining of
loops in the face.
After the intersection is complete, the outline of all the loops
in the face are drawn again so that the changes are visible.
The display is updated each time.

The second display shows the process of evaluating the boolean
formula on the fully intersected and cut objects.
First all the faces are evaluated.
Faces to be retained do not change, but
faces that are not part of the result are demoted to wire loops.
This is indicated by the color of the edges changing to yellow.
Wire loops that are not retained are demoted to wire edges.
Wire edges that are not retained are demoted to lone vertices
and disappear from the display.
Lone vertices that are not retained are killed, and vanish from
the display.

Even for the boolean combination of two relatively simple overlapping solids,
quite a few face intersections can be computed.
However, on typical workstations it is not uncommon to see ten
intersections computed and the diagnostic wireframes drawn to the screen
each second.
This high speed animated review of the intersection and boolean evaluation
operations is very valuable for teaching users how the boolean
operations work.
It can also be beneficial when a user wants to understand
the exact process that created a final NMG shape.
This capability also served as a very valuable debugging tool,
often permitting an intuitive grasp of programming errors to be
rapidly acquired -- pouring through formatted dumps of thousands of
NMG data structures was so difficult that it was reserved as
the technique of last resort.

\eject
So far, the primary motivation for computing an approximate surface
representation for an object has been to drive some
post processing'' application.
For example, the {\em ev} command computes an approximate surface
representation for the purpose of creating wireframes and shaded
polygon renderings for the user to view inside the {\bf mged} editor.
Approximate surface representations are also used to provide
appropriate kinds of input files for existing analysis applications.
But, the facetization and boolean combination mechanism that has been
constructed is completely general, and can be used for other purposes
also.

There may be circumstances when a designer may wish to
take a collection of primitive solids, tessellate and combine them
into some faceted shape,
and then store that faceted shape as the finished design.
If the object of the design task is to create a faceted object,
then this can be a powerful way of accomplishing that task.
It can be accomplished with the {\em facetize} command:
\beginprose
mged> {\em facetize newsol oldsol}
mged> {\em facetize newreg oldreg}
\endprose

The {\em facetize} command takes either a single pre-existing solid,
or a single pre-existing combination of solids (such as a group or region),
tessellates them according to the current tolerance settings,
and creates a new solid in the database that is the facetized
result, represented using the NMG data structures.
This newly created solid has exactly the same standing as
any other primitive solid:  it can be ray-traced, instanced,
or combined with other solids to create new shapes.
However, it is important to note that no record is made of
how this new solid was formed.
Thus, when the parameters of one of the
original solids inside the original region {\em oldreg}
is modified, this does {\em not} result in any changes being
propagated to the facetized version {\em newreg}.
If this effect is desired, it is necessary to delete {\em newreg}
and re-execute the {\em facetize} command, e.g.:
\beginprose
mged> {\em kill newreg}
mged> {\em facetize newreg oldreg}
\endprose

The same procedure must be followed if a different tolerance
for the facetization is desired.
% $Header: /vld/mike/talks/90nmg/RCS/joined.html,v 1.1 1998/05/01 06:53:13 mike Exp$

The capability to produce
an explicit approximate representation of the surface
of any object stored in the geometry database exists as a general
capability.
Any application program that links to the library {\bf librt}
can make use of this capability,
at any time in the analysis process.
Furthermore, use of this capability can be simultaneously intermixed with
other forms of interrogation supported by the library,
so that an application might perform some operations using
the approximate surface description, and other operations
using ray-tracing.
The application will have no knowledge of the underlying primitives
used to describe the objects in the database,
nor will the application be aware that the library
creates the surface description by
extracting the objects from the geometry
database, tessellating them into NMG solids,
and then combining them via boolean formulas.

Prior to the existence of this polygon rendering capability,
all renderings of the geometry databases had to be performed
using an optical simulation program based on ray-tracing.
While ray-tracing can produce some very beautiful images,
it can require a non-trivial amount of processing time
to create the images.
For the purpose of visualizing the shape of an object,
an image with much lower quality than those produced by ray-tracing
would be entirely acceptable,
if they could be produced in significantly less time.

The most immediate application of the explicit
surface representation is for visualization purposes:
making an optical image or {\em rendering} of objects
in the geometric database.
These images can be useful for a variety of applications.
Within the geometry editor {\bf mged}, a high-speed rendering
of a polygonal approximation of a shape is very useful for
inspecting the model.
This permits the user to verify that the design is as it should be.
It is also a very powerful technique for conveying information

If the display hardware is fast enough,
this capability could provide an improved interface for
many geometry editing operations.
Presently, modifications to solids are performed using the
wireframe display.  The modifications are entered
either numerically
(by entering specific parameters), or interactively
(by modifying key parameters through cursor manipulations on the screen).
In both cases, it can occasionally be difficult to judge by eye
whether the shape created is the desired one.
Being able to edit the object in a (seemingly) solid,
rendered form would greatly assist in this task.

Along the spectrum of rendering quality, there is a point
midway between the low quality of simple hardware polygon rendering,
and the high quality of ray-traced images.
This intermediate level of quality can be obtained using
software-based polygon rendering algorithms.
These algorithms tend to run faster than rendering algorithms
based on ray-tracing.
Especially when creating animation sequences that require
the rendering of hundreds or thousands of images,
having the ability to make this quality {\em vs.} speed tradeoff
can be a real boon.
This opportunity now exists.
An approximate surface description,
tessellated with appropriate tolerances, can be converted into
polygonal form and passed to existing polygonal rendering software.

In the design of vehicles,
it is very useful to be able to make predictions
about the thermal behavior of the vehicle
before the prototype is constructed.
This is important for ensuring passenger comfort
and proper cooling of temperature sensitive components.
In military applications, the patterns of heat radiation
are also quite important.
If simulation of the thermal behavior of the vehicle
reveals heat distribution that is not consistent with
the design criteria,
it is a simple matter to modify the vehicle geometry or
substitute different construction materials in an attempt
to improve the situation.
Re-running the thermal simulation will assess the effect
of these changes.

To simulate the thermal behavior of the vehicle,
it is necessary to calculate a complete heat
budget for the entire vehicle,
in addition to obtaining the geometry and material property information.
The heat budget must account for all the
internal sources of heat
such as engines, bearings, and road wheels,
such as cooling fins, air vents, and surface area skin'' in contact
with the open air.
The simulation must also
solar radiation and contact with the surface of the earth.

Accurate predictive thermal modeling is possible based on solid modeling,
using three dimensional finite element mesh (FEM) techniques.
The geometry is subdivided into small isomorphic solid volumes
or {\em nodes}, and the thermal properties of the material of each node are
recorded.
Heat flow is calculated between all node pairs,
mesh elements pairs are characterized by thermal coupling coefficients.
%%[Reference?]

However, a full three dimensional thermal model requires that a great
deal of material property information be known to a reasonable degree
of accuracy.
In many cases it is possible to compute a reasonable approximation
to the thermal behavior of a vehicle using a description of the surfaces
of the the vehicle, and some lump parameters'' for the
thermal mass of the various components, and for the primary
heat sources.
This approach, based on a geometry of flat plates,
is taken by
the Physically Reasonable Infra-red Signature Model (PRISM)
developed at the Keweenaw Research Center [REYN89].
Having the capability for generating an approximate surface
description of a BRL-CAD model will permit vehicle designs
stored in a BRL-CAD geometry database to be converted to a
form suitable for analysis by the PRISM application program.

Once the thermal behavior of the vehicle is predicted,
passenger comfort considerations can be directly assessed,
and no further processing is required.
For predicting a thermal signature, such as might be
seen on an imaging infra-red sensor, it is necessary to
take the simulated patterns of thermal energy radiation
and convolve them with the transfer function of the atmosphere
that lies between the vehicle and the sensor to
determine the patterns of energy presented to the sensor.
That pattern in turn must be convolved with the transfer function
of the sensor itself, in order to predict the signal
measured by the sensor.
In the design of sensor systems,
it is important to know the
nature of vehicle signatures over a range of detection bands
and for a variety of signal-processing schemes [RAPP76, RAPP83].
a method for computing this information as part of the
design loop.
This linkage provides opportunities for vehicle designers to
retain control over the thermal signatures of their vehicles,
as well as giving sensor designers an environment for
testing and refining improved sensors.

When a metallic vehicle is illuminated with radar energy,
that energy is partly absorbed,
and partly dispersed back into the surroundings.
Some of the illumination energy returns to the transmission
position, and it does so carrying an electronic signature''
of the vehicle [TOOM82].
Depending on the applications envisioned,
a vehicle designer is usually interested in either maximizing the
strength of the radar signature (for example, to make
boats and commercial aircraft easier to locate in foul weather),
or minimizing the strength and recognizability of the
radar signature, such as in the design of low-observable (stealth'')
aircraft.

A variety of different techniques exist to calculate the predicted
radar signature of a given vehicle.
The algorithms based on ray-tracing tend to handle multi-bounce
effects very well but are unable to simulate edge diffraction
and creeping wave phenomena.
However, algorithms based on feature-based
descriptions of the the vehicle or coarse polygonalizations
tend to handle diffraction and creeping waves acceptably,
but are unable to handle multiple bounce effects.
The best known technique for the simulation of radar signatures
is the Method-of-Moments technique [HARR82, MOOR84], which
requires a polygonalization of the surface of the vehicle
as input.

The nature of the calculation is much like that employed for
predicting heat flow, as outlined earlier.
However, in order to achieve high accuracy, the Method-of-Moment technique
requires that each surface polygon be no wider than one fifth
of one wavelength of the radar signal.
The relationship between frequency $f$ and wavelength $\lambda$ is given by
$$\lambda = { c \over f } = { { 3 \times 10^8 ~~ m/s } \over { f ~~ Hz } }$$

Radar frequencies begin in the UHF range
with P-band radars transmitting from 225 MHz to 390 MHz,
at a wavelength of about one meter [IEEE76].
A millimeter wave (W-band) radar transmitting at 94 GHz emits a signal with
a wavelength of 3{.}2 millimeters.
A radar transmitting at a higher frequency would have
a very short wavelength indeed.
The relationship between frequency $f$, wavelength $\lambda$, and
the maximum facet size
is given in Table 4.
Thus, the method of moments technique requires exceptionally fine
surface tessellations to be used.
Tessellating full size vehicles this finely produces a gargantuan
number of facets.
Computing a solution to problems of such size
is barely within the reach of present-day supercomputers.

\vbox{
\vskip 5pt
\centerline {
\vbox{ \offinterlineskip
\hrule
\halign{&\vrule#&\strut\hfil#\hfil& \hfil#& \hfil#& \hfil#&\vrule#\cr
&Band	& Frequency	& Wavelength	& Facet Size	&\cr\noalign{\hrule}
& P	& 300 MHz	& 1000 mm	& 600mm	&\cr
& L	& 1 GHz		& 300 mm	& 60mm	& \cr
& X	& 10 GHz	& 30 mm		& 6mm	&\cr
& W	& 94 GHz	& 3.2 mm	& \ 0.64mm	&\cr}
\hrule \vskip 5pt} %vbox end
} % centerline end

\centerline{Table 4 -- Relationship Between Frequency and Facet Size}
}

There are many reasons why the geometric database for a given design
might need to be transferred from the CAD system which originated it
to a completely different kind of CAD system.
This might be done because two different organizations use different
software packages
and they need to exchange design files.
Similarly,  as part of a comparison of the advantages of
different CAD systems, or of different analysis codes,
it would be necessary to import a reference design.
Or, some aspects of a design might be best handled
using software uniquely suited to a particular specialized task, such as
numerically-controlled (NC) machinery toolpath generation,
or ISO-compliant blueprint generation.

Having NMG objects as first class citizens'' in the geometry
database makes the task of importing faceted geometry from other
Existing faceted objects
either with no explicit topology, or
including explicit topology (most commonly using a winged edge
representation), can be easily converted into an NMG object
with full generality and no loss of topological information.
There are a great many systems that employ faceted representations,
so this is a significant capability.

Being able to convert models described using boolean combinations of
the rich set of primitive solids into explicit surface descriptions
also enables transfer of geometry in the outward direction as well.
Shapes that are originally modeled using the existing CSG database
% $Header: /vld/mike/talks/90nmg/RCS/joined.html,v 1.1 1998/05/01 06:53:13 mike Exp$

\vskip -22pt
%
At present, the only way to create an NMG object is via the
procedural interface of {\bf librt},
either from the {\bf mged} {\em facetize} command,
or from an outboard database converter or procedural
database generator that uses {\bf libwdb},
the library for writing databases.
Just as it is possible to edit all the other primitive solids,
it seems reasonable to permit users to edit NMG solids.
In principle, a basic NMG editing capability should not
be much different than the existing ARB editing capability,
which includes the ability to move vertices, move edges,
and move faces.
In addition to these familiar editing features,
some kind of interface to the Euller operators will need to be created,
so that new topological elements can be created,
existing elements can be killed,
and sets of existing elements can be joined together or split apart.

If the NMG editing capability is implemented in a very general way,
it should be possible to replace most of the existing faceted
object editing capabilities in {\bf mged} with interfaces to
a subroutine that implements editing of NMG objects under a
set of general constraints.
For example, when editing an ARB, if an edge is split,
all faces sharing that edge need to be split as well.
When moving a vertex, it must satisfy the constraint of
keeping planar all the faces that contain it.
A simple primitive shape like an ARB4 could be modified and
expanded into a much more complicated topological arrangement,
without forcing the user to destroy the original solid,
and re-create the shape using a more general primitive.

Editing B-spline control meshes could also be implemented using
this general NMG editing capability.
In this case, if an existing edge was to be split,
in order to create an extra control point,
a whole row or column of extra control points would have to be
inserted, to maintain the logically rectangular topology of
the B-spline control mesh.

When an NMG editing operation would complete, a series of simple
checks would see if the topology of the new shape qualified as
an existing primitive shape, such as an ARB8 or an ARBN.
If so, the new shape would be stored as the simpler and less general primitive,
so as to take advantage of the more efficient data storage and
faster analysis processing speeds inherrent in the less general
primitive shapes.

This capability should replace a large portion of the existing
geometry editing interface, and should provide geometry builders
with unparalleled flexibility and power in creating new shapes.

One of the most exciting current research projects at BRL is the
extension of the NMG framework to permit faces either to be planar
N-gons, or trimmed non-uniform rational B-splines (trimmed NURBS'').
This will permit many of the tessellation operations to be implemented
exactly, rather than as approximations.
This will also permit solids to enjoy the economy of having
most faces be represented as planar N-gons, which are very compact
and efficient to process, while those few faces that require sculptured
surface shape control can be represented as trimmed NURBS.
This combination provides both efficiency and full shape control
in the rich non-Manifold topological framework:
a combination that does not exist in any current commercial CAD system.

To date, most BRL-CAD applications programs have been
because of ray-tracing having a lengthy head start.
By choosing the ray sampling density
within the Nyquist limit
for a given spatial resolution,
many applications based on ray-tracing are well satisfied by
extracting ray/geometry intersection information.
However, a mathematical ray has as its cross section a point,
while physical objects have significant cross-sectional area.
This lack of cross-sectional area will always
Applications which simulate
particles or small rocks approaching the model might benefit
from having a direct cylinder/geometry intersection
capability, and applications which shine beams of light on the model
such as spotlights or
even highly collimated light such as laser light might benefit from
cone/geometry intersection capabilities [AMAN84, KIRK86].
Applications which are attempting to simulate wave effects might
be well expressed in terms of plane/geometry intersection curves,
and structural analysis routines would probably prefer to obtain
the geometry as a collection of connected hyperpatches.

While recent research has begun to explore techniques for
intersecting cylinders, cones, and planes with geometry [KAJI83],
ray-tracing and polygon-based techniques are
by far the most well developed approaches.
However, there are several additional types of interface
to the model geometry that are likely to be of
general applicability.

Many forms of energy flow analysis, such as
heat flow, vibrational analysis (acoustic energy flow),
and stress analysis require the use of 3-D Finite-Element Mesh (FEM)
techniques.
While there has been some work on using the ray-tracing paradigm
to construct finite element and finite difference meshes [LAGU89],
it has been difficult to deal with high spatial frequency
(fine detail) portions of the model.
In particular, meshing small diameter pipes is problematic,
since undersampling can cause the pipe to incorrectly be separated into
multiple pieces.
In order to improve on the current state of affairs,
it seems necessary to provide support for the generation of volume meshes
directly as part of the application interface.
This would provide the meshing algorithm to have unrestricted
and other internal data in order to perform a better job.

Even more promising still would be a strategy that takes advantage
of the NMG support.  A first pass might tessellate the model and
evaluate the booleans to produce a surface mesh.
The second pass would then take the surface mesh and fill the
interior (or exterior) volumes with appropriately chosen volume elements.
A very good fit could probably be achieved using only
parallelepiped (brick'') elements and 20-node superelements''.
The brick elements would be used to fill interior volume that
does not border on a face, and the superelements would be used for
volume that contacts a face.
Recourse could be made back to the underlying geometry
(perhaps via firing a few well chosen rays)
to get the curvature of the superelement faces to match the
curvature of the underlying primitive, rather than having to
rely strictly on the NMG planar-face approximation.

When support for trimmed NURBS faces has been added to the NMG
capability, it will be possible to represent all existing primitives
either with exact rational B-spline versions, or with very good
rational B-spline approximations.
This could be done even for faces that were completely planar.

This offers the hope that it might be possible (albeit memory intensive)
to convert an entire CSG solid model into a homogeneous collection of
non-uniform rational B-spline faces organized in a non-manifold
topological data structure.
In addition to the conceptual simplicity afforded by having a uniform
representation for shape,
this affords the opportunity to create new analysis codes that
can process curved surfaces, yet at least initially only have to
deal with one kind of shape.
This would also provide a very direct and natural interface
to spline based [ROGE90]
and Bezier-patch [BEZI74]
based modeling systems.

Given a homogeneous geometric representation such as the Trimmed
B-Splines just discussed which also has an analytic representation, a
further processing capability arises. Rather than interrogating the data
base by means sampling or subdivision techniques, the
direct mathematical manipulation of the source geometry through its
parametric representation becomes possible.
Calculations of physical properties requiring integration
over a surface can often be evaluated with greater accuracy
using an explicit analytic calculation than would be provided
by numerical evaluation. While this may be difficult
in general due to the complexity of the parametric expression,
some classes of surface representations good candidates.
Splines, for example, are
piecewise-polynomial functions which have relatively simple Fourier
transform representations. Since 2-D spatial Fourier transforms arise
frequently in far-field electromagnetic scattering calculations, exploitation
of the parametric spline representation is of interest in
predictive scattering calculations.

With the rapidly developing potential of symbolic calculation, treatment
of seemingly impossible formulas resulting from the geometry/physics
interaction may become tenable. This could help to reduce the trend towards
employing numerical methods at the onset of a problem and avoid
some of the accompanying instabilities and errors.

In this paper, a brief history of solid modeling has been
presented, with special emphasis placed on the central
role that the solid model plays in the design and analysis cycle.
A detailed look was taken at how different analysis applications
can interrogate the solid model to extract relevant information.
For CSG-Rep solid modeling system, the lack of an explicit representation
for the final, developed shapes was identified as a critical lack.

The systems engineering issues associated with
creating an explicit representation for CSG solid models
without loosing the fidelity of existing geometric databases
were considered, and a strategy using Non-Manifold Geometry
The implementation of the NMG data structures and algorithms
were presented in significant detail.
To convert existing primitive shapes into NMG objects,
the details of several tessellation algorithms were examined,
with particular attention being paid to the topic of rigorous
user-controlled error bounds on the algorithms.

Evaluating boolean combinations of tessellated objects has only
recently become tractable, since the use of the NMG data structures
makes these operations much more straightforward.
The details of a three-stage algorithm for boolean evaluation
were presented in sufficient detail to permit the average reader
to be able to implement this technique.

Finally, the existing user interface was presented, as well
as an overview of a whole gamut of new applications that
will now be able to process existing CSG geometry databases.
This represents a major and important capability,
the true significance of which will only become apparent
over the next few years.

The authors would like to thank Dr. Paul Deitz for
providing unflagging support and encouragement for this effort.
He has established a research atmosphere that breeds good work,
and is fun to work in.
The authors would also like to thank Prof. Dave Rogers for once again
persuading us to take the time to write all this down.
Finally, the clarity of the paper was greatly improved thanks to
numerous suggestions by Susanne Muuss and Christopher Johnson.
% WARNING:  This bibliography was created by hand from TROFF/REFER output.
% Don't just casually blow it away, edit it!
% (All the TeX standard bibliography packages are rotten).
%
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%
\beginitemlist
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P. H. Deitz,
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P. H. Deitz,
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P. H. Deitz,
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P. Deitz, W. Mermagen Jr, and P. Stay,
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Michael T. Goodrich,
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H. Gouraud,
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R. F. Harrington,
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J. T. Kajiya,
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G. Kedem,
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San Francisco CA, July 23, 1985.

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G. Kedem and J. L. Ellis,
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D. B. Kirk,
The Simulation of Natural Features Using Cone Tracing,''
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G. Laguna,
{\em Recent Advances in 3D Finite Difference Mesh Generation Using the BRL-CAD Package},
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David H. Laidlaw, W. Benjamin Trumbore, and John F. \break Hughes,
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J. H. Laning and S. J. Madden,
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MAGI,
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Zsuzsanna Molnar,
Advanced Engineering/Scientific Graphic Workstations,''
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J. Moore and R. Pizer (eds),
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Wiley, New York, 1984.

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M. J. Muuss,
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M. J. Muuss and P. Dykstra, K. Applin, G. Moss, E. Davisson, P. Stay, C. Kennedy,
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M. J. Muuss and P. Dykstra, K. Applin, G. Moss, P. Stay, C. Kennedy,
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A Solid Modeling System and Ray-Tracing Benchmark},
BRL Internal Publication, October 1988.

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M. J. Muuss,
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N. Okino and et al.,
{\em TIPS-1, '77 Version},
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J. R. Rapp,
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J. R. Rapp,
{\em A Computer Model for Estimating Infrared Sensor Response to Target and Background Thermal Emission Signatures},
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D. M. Ritchie and K. Thompson,
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S. W. Thomas,
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J. C. Toomay,
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Kevin J. Weiler,
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Kevin J. Weiler,
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\enditemlist

`