5.3 Representation Schemes
5.3.4 Euler Operations
Representation schemes based on using Euler operations to build objects are an attempt to have a boundary representation meet at least part of the validity issue head on. The idea is to permit only boundary representations that have a valid Euler char-acteristic. If we only allow operations that preserve the Euler characteristic or that change it in a well-defined way (such operations are called Euler operations), then we achieve this. Of course this is only a part of what is involved for an object not to be a nonsense object. Nevertheless we have at least preserved the combinatorial validity since the Euler characteristic is a basic invariant of combinatorial topology. As for metric validity, one still must do a careful analysis of face/face intersections. In any case, to say that a modeler is built on Euler operations means that it represents objects as a sequence of Euler operations.
Topologically, Euler operations are based on elementary collapses and expansions and/or cutting and pasting (see Sections 7.2.4 and 6.4 in [AgoM05], respectively).
Figure 5.13 shows two elementary collapse and expansion examples. One says that the space Yconsisting of the two segments on the right of Figure 5.13(a) is obtained from the solid triangle Xon the left via an elementary collapse of the cell cfrom the edgee. Conversely, the space Xis said to be obtained from Yvia an elementary expan-sion. Figure 5.13(b) shows another elementary collapse and expansion, this time involving a three-dimensional cell ABCDand a face ABC. Figure 5.14 shows a cutting
Figure 5.13. Elementary collapses/expansions.
and pasting example. Specifically, we show how to cut the torus to get a rectangle and how, looking at it backward, we can get the torus from the rectangle by pasting appro-priate edges together.
Elementary collapses or expansions do not change the Euler characteristic of a space. On the other hand, cutting and pasting operations usually do change the Euler characteristic. It turns out that these four operations do an excellent job to completely describe and define surfaces. (In higher dimensions things get more complicated.) Every surface, and hence solid in 3-space, can be obtained from a point by a sequence of elementary expansions, cuts, and pastes. Modelers based on Euler operations use a boundary representation for solids and simply define procedures that mimic the col-lapses, expansions, cutting, and pasting operations just described by modifying the cell structure of this boundary representation in a well-defined way.
Definition. TheEuler operation representationof polyhedra is defined by the collec-tion of pairs (X,(s1,s2, . . . , sk)), where Xis a polyhedra and s1,s2, . . . , skis a sequence of Euler operations that produces ∂Xstarting with the empty set.
Euler operations were first introduced by Baumgart in his thesis and then used in his computer vision program GEOMED ([Baum75]). Braid, Hillyard, and Stroud ([BrHS80]) showed that only five operators are needed to describe the boundary sur-faces of three-dimensional solids. Such a surface satisfies the Euler equation
where
They used a set of these Euler operations in their BUILD modeling system. Although one can make other choices for the five primitive operators, it seems that the bound-ary representation part of modelers built on Euler operations tend to use either
V = the number of vertices, E = the number of edges, F = the number of faces,
S = the number of solid components, and H = the number of holes in the solid.
V E F- + =2(S H- ),
Figure 5.14. Cutting and pasting.
Baumgart’s winged edge representation (see Section 5.8.1) or some variant of it, so that this is what these operators modify.
Historically, Euler operators were given cryptic mnemonic names consisting of letters. A few of these are shown below along with their meanings:
Using that notation, three typical operators were:
Figure 5.15 shows how one could create a solid tetrahedron using these operators.
The operators create the appropriate new data structure consisting of vertices, edges, faces, and solids and merge it into the current data structure. Along with each Euler operator that creates vertices, edges, or faces, there are operators that delete or kill them. This enables one to easily undo operations, a very desirable feature for a modeler.
There are good references for implementing modelers based on Euler operations.
One is the book by Mäntylä ([Mant88]), which describes a modeling program GWB (the Geometric WorkBench). Another is the book by Chiyokura ([Chiy88]), which describes the modeling program DESIGNBASE. Euler operations were originally defined only for polyhedra but were extended to curved surfaces by Chiyokura.
To summarize, modelers based on Euler operations are really “ordinary” b-rep modelers except that the objects and boundary representations that can be built are constrained by the particular Euler operators that were chosen, so that they at least have combinatorial validity. The Euler operators are flexible enough though so that
MEV
-make ede and vertex MFE make face and edge
MBFV make body, face, and vertex M make K kill L - loop
V vertex E edge F face B body solid
-
-- - - S
-Figure 5.15. Building a tetrahedron with Euler operations.
Sweeps sometimes become inputs to other representations. For example, in CADD (a program developed by McDonnell Douglas) one can translate certain sweep repre-sentations, such as translational and rotational ones, into boundary representations.
Related to sweeps is the multiple sweeping operation using quaternions described in [HarA02]. There are also the generalized cylinders of Binford ([Binf71]). See Figure 5.19. Here the “sweeping” is parameterized. We shall now discuss a representation scheme developed by J. Snyder at Caltech that is more general yet. It was the basis for the GENMOD modeling system, which Snyder’s book [Snyd92] describes in great detail.
Definition. Agenerative modelis a shape generated by a continuous transformation of a shape called the generator.
Arbitrary transformations of the generator are allowed. There is no restriction as to the dimension of the model. The general form of a parameterization S(u,v) for a generative model which is a surface is
(5.3) whereg: [a,b] ÆR3is a curve in R3and f : R3¥RÆR3is an arbitrary function. One of the simplest examples of this is where one sweeps a circle along a straight line to get a cylinder. Specifically, let
be the standard parameterization of the unit circle. Define f by f(p,v)=p+(0 0, ,v).
g
p p
: ,
cos , sin , 0 1
2 2 0
[ ]Æ 3
Æ( )
R
u u u
S u v( , )=f(g( )u v, ,) Figure 5.18. Problem with sweeps.
Figure 5.19. Generalized cylinder.
Definition. The generative modeling representation consists of pairs (X,F), where F is a parameterization of the generative model Xof the form shown in equation (5.4).
The driving force behind GENMOD was correcting some perceived deficiencies in the geometric modeling systems of that time and some key defining points listed by [Snyd92] for the generative modeling approach as implemented in GENMOD are:
(1) The representation is a generalization of the sweep representation.
(2) Shapes are specified procedurally.
(3) Specifying a shape involves combining lower-dimensional shapes into higher-dimensional ones.
(4) An interactive shape description language allows low- and high-level opera-tors on parametric functions.
(5) It is closed, that is, the outputs to operations can be inputs to operations (like CSG).
(6) It allows parameterized shapes whose parameters a user can change.
(7) It supports powerful high-level operators and functions, such as reparameterizing a curve by arc length,
computing the volume of a shape enclosed by surface patches, and computing distances between shapes.
These operations are closed and free of approximation error.
(8) It supports deformation operators, CSG, and implicitly defined shapes.
(9) One has the ability to control the error in the representation.
A large variety of symbolic operators on the parameterizations and their coordi-nates help the user define generative models, such as vector and matrix operations, differentiation (partial derivatives), integration, concatenation, and constraint opera-tors. Since parameterizations can be thought of as vector fields, another useful oper-ator is one that solves ordinary differential equations. GENMOD had a language in which a user could define models using the various operators.
Now, models will have to be displayed. By converting to polygonal meshes and ad hoc error control, the interactive rendering of generative models becomes feasible.
One can specify the subdivisions in two ways: uniform in domain or adaptive sam-pling. More realistic images can be obtained at the expense of speed.
For accuracy, GENMOD used interval analysis. Interval analysis (see Chapter 18) is an attempt to make numeric computations on a computer more robust and has its advantages and disadvantages. Snyder argued for its use in geometric modeling and described various applications to computing nonintersecting boundaries of offset curves and surfaces, approximating implicitly defined curves and surfaces, and trimmed surfaces and CSG operations on them.
In summary, three more advantages used by Snyder to justify the generative modeling approach are:
(1) The representation handles all dimensions, is high-level, and extensible.
(2) Using a high-level interpreted language, the mathematically knowledgeable user can easily build a library of useful shapes.
(3) An adequate number of robust tools for rendering and manipulating genera-tive models exist.