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Computer Graphics, 26, 2, July 1992 Topological Design of Sculptured Surfaces Helaman Ferguson Supercomputing Research Center Alyn Rockwood Computer Science Dept. Arizona State University Jordan Coxt Dept. of Mech. Eng. Purdue University Bowie, MD Tempe, AZ West Lafayette, I&J Topology is primal geometry. Our design philosophy embodies this principle. We report on a new surface &sign perspective based on a “marked” polygon for each object. The marked polygon captures the topology of the object surface. We construct multiply periodic mappings from polygon to sculptured surface. The mappings arise naturally from the topology and other design considerations. Hence we give a single domain global parameteriration for surfaces with handles. Examples demonstrate the design of sculptured objects and their ntanufimture. boundaries. The burden of maintaining topological integrity falls to the designer. In these methods, there is no single parameter space, there are many sets of separate coordinate functions. Control of the topology may be simple for some shapes, but it is difficult for topologically complex ones. Some “solid” modelling systems check topology after the design stage, e.g. via the Euler-Poincark formulae mof89]. The check only determines when an invalid operation has occured. but does not participate in the design process. This paper describes a design philosophy that includes surface topology as an integral part. Our interest in this subject came from a desire to automate the sculpture of mathematical concepts such as in Figures 1.1 and 1.2 (see [Ferf39, Fer90, or Roc861). CR Categories: 1.3.5 [Computer Graphics] Computational Geometry and Object Modeling - Geometric Algorithms; 1.3.6 [Computer Graphics] Methodologies and Techniques - Interactive Techniques. Additional Key Words and Phrases: Computer-aided design, sculptured surfaces, topology, marked polygon, automorphic functions, multiply periodic functions, boundary value problems. 1. Introduction Creating shape excites many artistic and scientific minds. It is important to areas as diverse as sculpture, mathematics, cartoon animation, molecular modeling, architecture, mechanical design. It is also one of the most difficult to automate because it demands a comprehensive toolset to handle such needs as topology, geometry, analysis, and manufacture. Traditional design methods (see [Far88, Hof891) define the surface of an object as a quiltwork of many patches; topological issues are treated superficially during the design stage. For example, the methods of Coons and BBzier and the B-rep method of solid modeling define an object as a collection of patches (or faces) that match at tcurrently at the Dept of Mech. Eng., Brigham Young University, Provo, UT. Permission IO copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage. the ACM copyright notice and the title of the publication and its date appear. and notice is given that copying is by permission of the Associati?m for Computing Machinery. To copy otherwise, or to repubhbh. requires a fee and/or specific permission. Figure 1.1. Automatically sculptured “Umbilic Torus NC.” The continuous NC path was generated by a Peano- Hilbert surface filling curve. The torus itself is defined on a single domain. 1” 1992 ACM-0-X9791-479-1/92100710149 $01.50 149
Transcript
Page 1: Topological Design of Sculptured Surfacespapers.cumincad.org/data/works/att/2b7a.content.pdf · 2004-04-06 · design, sculptured surfaces, topology, marked polygon, automorphic functions,

Computer Graphics, 26, 2, July 1992

Topological Design of Sculptured Surfaces

Helaman FergusonSupercomputing Research

Center

Alyn RockwoodComputer Science Dept.Arizona State University

Jordan CoxtDept. of Mech. Eng.Purdue University

Bowie, MD Tempe, AZ West Lafayette, I&J

Topology is primal geometry. Our designphilosophy embodies this principle. We report on a newsurface &sign perspective based on a “marked” polygon foreach object. The marked polygon captures the topology ofthe object surface. We construct multiply periodicmappings from polygon to sculptured surface. Themappings arise naturally from the topology and other designconsiderations. Hence we give a single domain globalparameteriration for surfaces with handles. Examplesdemonstrate the design of sculptured objects and theirntanufimture.

boundaries. The burden of maintaining topological integrityfalls to the designer. In these methods, there is no singleparameter space, there are many sets of separate coordinatefunctions. Control of the topology may be simple for someshapes, but it is difficult for topologically complex ones.Some “solid” modelling systems check topology after thedesign stage, e.g. via the Euler-Poincark formulae mof89].The check only determines when an invalid operation hasoccured. but does not participate in the design process.

This paper describes a design philosophy that includessurface topology as an integral part. Our interest in thissubject came from a desire to automate the sculpture ofmathematical concepts such as in Figures 1.1 and 1.2 (see[Ferf39, Fer90, or Roc861).

CR Categories: 1.3.5 [Computer Graphics]Computational Geometry and Object Modeling - GeometricAlgorithms; 1.3.6 [Computer Graphics] Methodologies andTechniques - Interactive Techniques.Additional Key Words and Phrases: Computer-aideddesign, sculptured surfaces, topology, marked polygon,automorphic functions, multiply periodic functions,boundary value problems.

1. Introduction

Creating shape excites many artistic and scientificminds. It is important to areas as diverse as sculpture,mathematics, cartoon animation, molecular modeling,architecture, mechanical design. It is also one of the mostdifficult to automate because it demands a comprehensivetoolset to handle such needs as topology, geometry,analysis, and manufacture.

Traditional design methods (see [Far88, Hof891)define the surface of an object as a quiltwork of manypatches; topological issues are treated superficially duringthe design stage. For example, the methods of Coons andBBzier and the B-rep method of solid modeling define anobject as a collection of patches (or faces) that match at

tcurrently at the Dept of Mech. Eng., Brigham YoungUniversity, Provo, UT.

Permission IO copy without fee all or part of this material is grantedprovided that the copies are not made or distributed for directcommercial advantage. the ACM copyright notice and the title of thepublication and its date appear. and notice is given that copying is bypermission of the Associati?m for Computing Machinery. To copyotherwise, or to repubhbh. requires a fee and/or specific permission.

Figure 1.1. Automatically sculptured “Umbilic TorusNC.” The continuous NC path was generated by a Peano-Hilbert surface filling curve. The torus itself is defined on a

single domain.

1” 1992 ACM-0-X9791-479-1/92100710149 $01.50 149

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SIGGRAPH ‘92 Chicago, July 26-31, 1992

Figure 1.2. Milling the “Umbilic Torus NC” on a threeaxis milling machine, ball end mill tangent to the torussurface defined globally over a single parameuic patch.

Our approach decomposes the surface of an objectinto a polygon, using a series of virtual surface cuts. Thevirtual cuts are identified with oriented markings that boundthe polygon. Such a marked polygon encodes the topologyof the object. A mapping from the two dimensional interiorof this polygon into three dimensional space is thenconstructed. The domain of the mapping is a the interior ofthe polygon; the image of the mapping is the surface of theobject. The surface is a single topologically consistentpatch.

As we investigated the problem, the value of thesingle polygon patch approach for analysis and manufacturewas better appreciated. Even topologically simple objectsbenefit. The texture on the object in Figure 1 depends on apattern that is continuous across the marked boundaries ofthe domain and drove the milling tool.

In his famous Erlangen lecture, [Fir82], Felix Kleindescribed geometries by their associated mappings,especially groups of mappings. Objects within thegeometries are related and classified by the transformationsbetween them. Rigid body transformations preservecongruence, projective transformations preserve similarity,Cl transformations preserve differentiability, C? preservestopological structure. For example, a cuboid with comersis C” to a sphere but not Cl; a sphere is not Cc to a torus.

The most fundamental mappings are C?, these preservetopolosY *

Topology is primal geometry. Our design philosophyembodies this principle.

This paper is organized as follows. Section 2introduces the topological concepts mapping, domain,image, homeomorphism, immersion, embedding, and genus.Section 3 gives three examples from this design philosophy:a circle, a torus, and a genus three surface with many Beziersub-patches over a single twelve-sided polygon. Section 4reveals how to give a single polygon parameterization of asurface of arbitrary genus by constructing real analytic(smooth) multiply-periodic coordinate functions. Section 5constructs an embedding of a double torus by solvingboundary value problems.

2. Surface Topology

Figuratively, two objects are homeomorphic if oneobject can be re-formed into the other by a continuousdeformation that does not tear the object nor make it self-intersect. For a readable exposition see [Nas831, forrigorous definitions see [Ho&l]. We keep to an intuitivelevel.

Consider a polygon in the plane. Think of thepolygon as elastic material to be sewn up along the edges.Give each edge a direction and name each edge with a captialletter. We define, figuratively, a surface to be a ‘sewing up’of this polygon in space by stitching together the edges withthe same capital letters. Assume there are no leftover edgesand that the sewing is direct, without twisting or knottingthe material. There are two possibilities: the sewn materialmay self-intersect or not. If there are self-intersections wecall the spatial result an immersion. If there are none wecall the resulting surface S an embedding. The embeddingsbound natural objects.

There exists a set of directed virtual cuts on anysurface S that induce a one-to-one correspondance from thesurface with virtual cuts to its polygon with directed edges(see [Fir82, Sie88]). The polygon together with its directedand labelled edges is called the marked polygon of thesurface. Note that a pairwise identification of two edgesoccurs from a single cut in the surface. This pair of edges isidentified by the same letter, but differentiated by thesuperscripts. Figure 2.1 illustrates two cuts which convert atorus into a marked rectangle. It is also the topology usedto create the “Umbilic Torus NC” of Figure 1.1.

Figure 2.1. The torus and its fundamental polygon with formal word ABA-lB-1.

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Move clockwise around the polygon and give theletter a “-1” superscript if its edge runs counter to themovement, otherwise the letter gets no superscript. The“word”that describes the toms in Figure 2.1 is ABA- lB-l.The word ABAB-l, for which the direction of the thirdedgediffers from the torus, represents the Klein bottle and isalways an immersion. Immersions and embedding are bothpossible for tori.

A surface is orientable if an inside and an outside canbe distinguished, i.e. bounds a natural object. The sphereand torus are orientable; the Klein bottle is not. Thefollowing theorem classifies all orientable surfaces:

A closed, orientable surface has a marked polygon wordeither of the form AA-l or AI BIA1-l B1-l ... AgBgAg-l

Bg-l (g> 0).

The fmt case is a sphere. The second can be thoughtof as gluing g tori together. The number g is the genus ofthe object. It counts the number of “donut holes.” Theseare called handles. All objects of the same genus aretopologically equivalen~ they can be continuously deformedfrom one to another. A corollary to the above theorem is

A closed, orientable suflace (g>O) has a markedpolygon with 4g sides.

Many cutting schemes are possible for objects ofgenus g>l. The textbook cuts [Fu82, Sre88] usuallyemanate from a central point and emphasize symmetries asin Figure 6. We have devised non-standard cuts that arebetter for some purposes. Consider the double torus(“Borus”)shown in Figure 2.2.

All cuts begin at point A inside on the bottom of thelower handle. The fnt cut divides the bottom loop into twosleeves. The next one slits the sleeve orthogonally to thefust along C. The next pair of cuts run from one sleevecomer to the other throughalong B and then D.

The example is easily generalized to a genus g objecqg>2. The object is first deformed into an extended figure“8.” All cuts are made from the same point. The even cutstraverse through each handle and return to the point. Theodd set of cuts start from the point and pass through theantipodal point in the handle without intersecting any othercuts.

The next step in the marked polygon design processdetermines vector valued functions that define the surface.Consider the function fl P+S, for P a polygon in R2,surface S in R 3 and f: (u, v) +(x, y, z). Notice thatpairwise related edges of the marked polygon must map sotheirdirectionscoincide on the surface.

The function ~ is vector-valued with coordinatefunctions fj: R2 + R, j=l,2,3, where x = ~l(u,v), y =

\2(u,v) and z = ~3(u,v). These coordinate functions have the

B-l B

polygon as a single common domain. Figure 2.2. Illustrationof the systemof virtualcuts on adoubletorusas it k openedupto a stopsign shapedmarkedpolygon with formal word ACBDA-lC-lB-lD-l.

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SIGGRAPH ‘92 Chicago, July 26-31, 1992

3. The Circle, The Torus, and the Embeddingby Many Bkzier Patches.

3.1. The Circle. We now apply our designphilosophy to the simplest posssible example which has allthe essential elements of the general case -- a circle. The cuton a circle corresponds to a single point. The onedimensional polygon domain corresponds to a line segment.We take the line segment to be the interval (-l/2, l/21.Note that integer translations of this interval cover the realline. A polynomial mapping of the interval into the planewill image some kind of curve, which may cross itself(immersion) or be simple without self-intersection points(embedding). Two x and y coordinate polynomials whichgive a circle-like curve are x(t) = l-16r* and y(t)=f( 14*).But polynomials are not periodic and the curve has a comer.Its tear-drop shape is in Figure 3.1. The coordinatefunctions should have period 1 since the distance from cut tocut on the (-l/2,1/2] interval is 1.

Figure 3.1. Left: A polynomial embedding of a circle.Note the comer at the cut point of the circle, becausepolynomials are not periodic. Right: A periodic embeddingof a circle using the same polynomials in the periodizingtransformation.

How does one make numerically computable periodicfunctions out of polynomials? For any polynomial f(t), therapidly convergent sum

F(f)= c f(f+k)e-(‘+QZis periodic: F has period 1, F(z) = F(t+l), because the sum isover all integers Z. Replace f by x in the sum to get acoordinate function X(t) and f by y to get Y(f). Then X andY both have period 1. Normalizing X and Y appropriatelyand taking eleven terms in the sum gives the classicalperiodic cosine and sine functions within eleven decimalplaces, cf. Figure 3.1.

We recapitulate the features of this simple example:1) we began with an idea of what we wanted to draw, aclosed ‘circular’ curve which can be cut open at one point.2) we picked some qualitative guesses for the coordinatefunctions which could give an embedding. 3) We recognizedthe essential periodic nature of the curve and made ourguesses periodic to get a smooth curve. Recognizing aperiod for a circle is the same idea as recognizing thetopological genus or system of virtual cuts for a surface.The periods are roughly speaking distances between cuts.

3.2. The Torus. The next simplest case is thetorus. The torus is a Cartesian product of two circles, so wecould replicate our design philosophy in this similar

context: 1) We think of an inner tube shape, mark it withtwo virtual cuts so that the shape corresponds to a markedsquare. 2) We pick some qualitative bivariate polynomialguesses to give an embedding. 3) We recognize the torus asdoubly periodic and convert our guesses into doubly periodiccoordinate functions with a bivariate sum over all pairs ofintegers, Z x Z.

This leads to the following parametric equations for atorus with major radius a and minor radius b: (u,v) ->((a+bcos(2zv)) cos(2nu). (a+bcos(*xv)) sin(*xu), bsin(2ltv) ). This maps the marked polygon (-l/2,1/2] x(-l/2,1/2] to the torus. This mapping is an embedding ifa>b>o. Note that the coordinate functions in the vector aredoubly periodic, i.e., invariant under replacing u by u+l or vby v+l. Furthermore these doubly periodic functions arereal analytic (smooth) and bounded.

3.3. Traditional method. It is possible toformulate the coordinate functions using traditional designmethods, e.g. Bezier cross plots [Far88], which arepiecewise polynomial. The single twelve sided polygon forthe genus three surface in Figure 3.3 organizes the hundredor so Bt5zier patches in a coherent way.

In this case, the domain polygon is tessellated intofour-sided regions which are associated with the domains ofthe polynomial patches; thus inducing a topological aspectinto the design paradigm for conventional patch methods.

Figure 3.3. Genus 3 object from many BCzier patchescoherently organized over a marked 12-gon.

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Computer Graphics, 26, 2, July 1992

4. Defining the Embedding with MultiplyPeriodic Functions.

Natural sculptured objects are always bounded by asurface, are closed, and have genus. For a surface to havegenus amounts to having multiple periods. A simple closedcurve in space is periodic and has periodic coordimtefunctions. Closed surfaces in space can be defined bycoordinate functions thatare multiply periodic. A rich classof multiply periodic functions are available in the classicalliterature [For51, Sie88]. In this literature multiplyperiodic functions are usually called automo~hic functionsbecause these functions are invarient under a group ofautomorphisms. A problem with the classical literature isthat it tends to be preoccupied with complex analyticfunctions that do not give bounded embedding. Wecircumvent this obstruction by constructing enough realanalytic functions which are bounded. These real analytic(smooth) functions are the analogs of the classical doublyperiodic real analytic functions of variables u and v, thefinite Fourier series in Sin(u), Sin(v), Cos(u), Cos(v)encountered for the torus in Section 3.2. Another feature isthat the multiple periods of interest for surfaces of genusgreater than one are not additive. The periods are elementsof a multiplicative and non-commutative group oftransformationsof the variables u and v.

Multiply periodic functions can h of any arbitrarydifferentiability class. In what follows, we solve theproblem of defining a smooth embedding from the markedpolygon to the surface of an object. It allows the embeddingto be defined smoothly across the boundary marks of thedomain, and thus the virtual cuts of the object. We realizethis markedpolygon in the hyperbolic disc with sides whichare geodesics in the hyperbolic geometry. The multipleperiods will be hyperbolic translations. All these hyperbolictranslations form a multiplicative group, The hyperbolictranslationstessellate the hyperbolic disc with copies of themarkedpolygon. Traversingthe polygon from one tesselantto a neighbor is equivalent to recentering the originalpolygon at another point of the tmundary. A “periodizing”transformation is defined on the tessellated disc that takesfunctions, say polynomials, over the fundamental domainand blends them at the marks, making them seamless, i.e.infinitely smooth across the virtual cuts. This was done fortaking the tear drop to the smooth circle in Section 3.1.This “multiple periodizing” transformation involves a sumover all of the elements of the group of hyperbolictranslations. A different group will be defined for eachgenus.

4.1. The Hyperbolic Disc. The naturalgeometry for the surfaces with handles which are of interestto us is a hyperbolic geometry. A surface with g handlescan be cut with 2g cuts. The comesponding polygon has 4gsides and can be laid down in a hyperbolic geometry,mar83]. One model for this geometry, the Poincan5model,is given by imposing some geometric structureon the unitdisc (z I ZZ*< 1, z E C } where C is the usual complexnumber field and z* is the complex conjugate. The pointsof this hyperbolic geometry are the points of the unit disc.The lines or geodesics of this hyperbolic geometry are given

by arcs in the unit disc of circles peqwmdicularto the discboundary at their endpoints. Lengths of arc segments,angles between intersecting arcs, and area of polygonsbounded by arcs are all &fincd, [Sie66]. One-to-one, onto,analytic mappings of the disc to itself that pr-e lengthsand angles are given by linear fractional transformationsofthe form z -> (az+b)/(b*z+a*) where a, b are complexnumbers such that aa*-bb*=l. Among these are thehyperbolic rzanslationswe define for the periods of a spherewith finitely many handles.

4.2. The Single Marked Polygon Patch.As described in Section 2, it is possible to mark a surfacewith exactly g handles in such a way that when the surfacecan be ‘sewn up’ from the interior of a polygon of exactly4g sides, where opposite sides are identified and the sewingis done without twisting. We specify a particular suchpolygon with 4g circular arc sides in the hyperbolic disc.This special set of 4g circular arcs comprises the boundaryof the what we will call the canonical 4g -gon. It issymmetric about the origin. Ad@ining arcs are constrainedto meet a vertex with interior angles of exactly n/2gradians. Hence, the sum of these 4g angles is exactly 2x.This ensures, by a theorem of Poincar6 ~ir82], that a groupof h~rbolic translations, rg which we define Mow, of the4g-gon exactly covem the hyperbolic disc.

Each boundary circular arc is given by the inside arcof a chcle of radius

sin(7t/4g)r=‘e

centered at integer multiples of x/2g and radius cg=rgcot(n/4g). For genus two,

Q=J%”dc2=J%The mdhs of the excribed circle containing the eight

vertices of the poly on is 2-1/4 and the radius of thebinscribed circle is (21 - 1)12.

4.3. The Group rg of Multiple Periods.Define lg and mg by

$tan;(l-:)

and

21*m=—.

g 1+~

Define the hyperbolic translation

z-mTg:z+—,

1- m~z

of intinite order,and the single rotation

L -1

Rg:z+e2g “ z,

I 53

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SIGGRAPH ‘92 Chicago, July 26-31. 1992

of order 4s. Note that Tg and Rg do not commute. All ofthe elements of rg are generated by the 4g fundamentalhyperbolic translations

Hg,~Rg-k Ts Rsk, kO.1.2 ,..., 4g- 1.

4.4. Enumeration of the Elements of rg .Every period or element of rg is realized by a “word” inthe symbols Hg& of the form Hg&tal . . . H~J,,,~, wherethe ui are non-negative integers. The length of this word isgiven by the non-negative integer q+...+u,. These wordscorrespond to fractional linear transformations and there arerelations among the words; some words are equal to otherwords that are spelled differently, thii gives an equivalenceclass for a given word. A shortest word in a set ofequivalent words is called a geodesic word. It is possible towrite down finite state automata which generate all geodesicwords in increasing order IEps921. It is important toenumerate algorithmically the geodesic words, in order ofincreasing word length, to efftciently compute a large classof multiply periodic functions.

For the case of the double torus and the group r2,there are eight generators. Set H2.t = HL The generatorsare I-lo. Hl, Hz, H3, Hq. H5. Hg and H7, and the particularword HoH~H~H~H~H~H~H~ is equivalent to the identityword of zero length. The numbers of words of length 0.1,2, 3,4, 5 ,... are 1, 8, 56, 392. 2736, 19096 ,... respectively.The generating function for this double torus case is( 1+2t+2r2+2$+r4)/( l-6t-6r2-6t3+r4), [Can84,92].

fjer = (1 - 1~1’ )j Re(zk)

a

f.j L = (l- lzl’)’ Im(zk)

for j=2,3 ,... and kO,1,2 ,... . Note that the sum is over rs ;thus the need for the algorithm above to generate geodesicelements of this group in order. These series convergerapidly; for visual accuracy on the double torus fewer than1+8+56+392=457 terms suffice. See Figure 4.5 for anexample basis function.

Figure 4.5. A multiply periodic basis function for thegroup r2 plotted over the marked octagon in the hyperbolicdisc.

Figure 4.4. Images of the octagon for genus two in thehyperbolic disc under hyperbolic translations by the geodesicwords of the group r2. The center octagon is the identitysurrounded by the alphabet of generators Ho, Hl, Hz, H3,H4, H5, H6, H7, then words HiHj, then HiHjHk, . . . .

4.5. Multiply Periodic Functions for theSphere with g handles, g>l. A set of I’g multplyperiodic basis functions is given by

q,k(Z) = C fj,k(H * f)Her*

Where

Figure 4.6. An infinitely smooth double torus definableon a single marked octagon.

5. Defining the Embedding by SolvingBoundary Value Problems.

This approach to embedding homeomorphisms posesseparate boundary value problems for the x, y, and zcoordinate functions. The solution to these 2-dimensionalboundary value problems models the specific coordinatefunction over the entire domain. The 3-tuples are formed bycollecting the values of each coordinate function at specificparametric domain points. The parametric domain connectsthese 3-tuples to give the geometry of the de&d object..

Each boundary value problem is constructed byselecting an appropriate differential operator which modelsthe geometric shape of the coordinate function and then

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Computer Graphics, 26, 2. July 1992

applying boundary conditions derived from the appropriatecoordinate values of the markings on the actual object.Similar methods have been used to model surface patches(see [ Blo90, Blo9Ocl ). Since the markings represent theboundaries of the domain, see Figure 6, it is appropriate todefine the boundary conditions for the boundary valueproblem to be the coordinates at these markings.

The development of the solutions to the boundaryvalue problems can be accomplished through approximationtechniques like finite element methods (see Lap82). Forgenus g > 1 objects the domain is 4g-sided. These 4g sidesrequire that the solution approximation technique includenon-rectangles to “mesh” the domain. For this nextexample of a double torus, domain composition methodswere used (see [Cox9la, Cox9lb. Cox9lcl). These methodsallow overlaps in the finite elements. Thus the 4g-sideddomain can be meshed with overlaps.

The selection of the differential operator in the doubletorus example is based on a traditional potential problem.Potential problems (i.e. heat transfer, pressure, etc.) aremodeled using Laplace or Poisson operators. The differentialoperator choice should be dictated by the shape of the objectand the periodicity of the coordinate functions. Theselection of the differential operator is important and furtherwork on this topic is forthcoming. For this example theLaplace operator generally produces the required smoothnessin the coordinate function. Most existing finite elementpackages provide Laplace or Poisson operators.

Boundary conditions specified along cuts

Extrema of torus

Figure 6. Topological form of a double torus where theembedding is constructed by solving three boundary valueproblems. Boundary conditions are determined from thecoordinates along the four cycles or virtual cuts as marked.Internal constraints to force an embedding arise from theouter and inner radii.

Once the partial differential equation, the finiteelement mesh, and the complete set of boundary conditionsare prescribed, there will exist a unique solution. However,it may not produce the desired coordinate function. Internalconstraint conditions are added to the specification of theproblem to achieve the desired results. These constraintsspecify some of the coordinate x,y, or z values in theinterior of the solution surface. These can correspond toextrema of the object (i.e. the outer radius or inner radius ofthe double torus example.) Each of the coordinate boundaryvalue problems will have different internal constraints. The

internal constraints can be applied using penalty functions.Figure 7 shows the z-coordinate function for a specificembedding and decomposition of the double torus. Forexample, Figure 7 shows a cross shaped flat area above thecoordinate function indicating the location of the internalconstraints. These constraints force the solution to be zeroalong the corresponding curve on the surface. Theseconstraints prevent self-intersections in the double torus asthe octagonal domain is embedded in space. The ftrst sub-figure of Figure 7 shows the double torus. The second sub-figure shows the z-coordinate function as height over theoctagonal domain. Both tigures are contoured and renderedwith a height related color map

A 3-tuple for the embedding is constructed from thex, y, and z coordinate values at the same domain point. Allthe 3-tuples of the embedding thus come from the threeboundary value solutions representing the coordinatefunctions. If each of the coordinate functions are generatedusing equivalent finite element meshes in the domain, thenthe solutions at the nodes can be used directly as the 3-tuplesthat model the object. The connectivity of the elements inthe domain will translate into appropriate connectivity in theobject space. The double torus of Figure 7 is modeled bycombining each of the elements in the three coordinate mapsinto elements covering the object. This is the image of theembedding from the octagonal domain into three

Figure 7. The double torus showing views of the z-coordinate function.

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SIGGRAPH ’92 Chicago, July 26-31, 1992

The surface of the double torus of Figure 7 hasanomalous bumps and ridges. Changes in the partialdifferential operator affect bumpiness. For example, the“stiffness” of the surfacecould be increased, the order orform of the differential operatorcan be modi.tkl, or a bodyforce applied to smooth the resulting object. Furtherinvestigation into these methods is ongoing. Anotherproblem is the distortion of the grid on the octagonaldomain to the image grid on the double torus. More researchon partial differential operators and more &velopment ofmethods of selecting internalconstraints is invited to conmolthe modeling of coordinate functions. It is significant tonote that the genus of the object is preserved throughoutwhile them is great flexibility in the shape and locations ofthe handles. This gives a spatial richness needed forgeometric design.

6. Conclusion

Our approach is unique in its incorporation oftopology as well as geometry into the design process. Thisleads to more comprehensive models for scientificvisualization and manufacturingneeds and is well-suited to avariety of representations of complex objects which arisenaturally from tie abstrtwtionprocess in scientitlc research.A basic outstanding problem is to provide a good set ofsufficient conditions (on the coordinate functions) to give anembedding of a sphere with handles. The techniquesintroducedhere work in principle for any dimension. Futuredirections includes applying these ideas for volume domainsand volume images with attachedfunctions.

Acknowledgements

We appreciate Sareddy Madhukar of SpatialTechnologies, Boulder, CO for help on Figure 3.3 and DaveSmittley of the Super computing Research Center Bowie,MD for help with Figure 4.5.

References

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@lW89]

[can84]

[Can92]

[Cox91a]

Bloor, M. I. G., Wilson, M. J., “Using PartialDifferential Equations To Generate Free-FormSurfaces”, Computer-Ai&d Design, vol 22,number 1, May 1990.Bkmr, M. I. G., Wilson, M. J., “Generating N-sided Patches with Partial DifferentialEquations”, Computer Graphics International,Wyvil, B. (Editor), Springer-Verlag 1989.Cannon, J., “The Combinatorial Structure ofCocompact Discrete Hyperbolic Groups,nGeometrt”aeDedicata, VOI16,1984,123-148.Cannon, J,, Wagmich, P., “GrowthFunctions ofSurface Groups”, to appear in Math. Annalen,1992.Cox, J., “Domain Composition Methods ForCombining Geometric And Continuum FieldModels”, PhD Thesis, Purdue University, WestLafayette, IN, December 1991.

[Cox91b]

[COX91C]

[COX88]

rEPs921

rFar88]

l-Fe&9]

l-Fer90]

rFer92]

Fir82]

FO151]

rHof90]

[~2]

B-1

[Sie88]

Cox, J., Anderson, D., C., “Single ModelFormulations That Link Engineering AnalysisWith Geometric Modeling”, Product Modelingfor Computer-Ai&d Design and Man#actura”ng,J, Turner, J. Pegna and M. Wozny (Editors),Elsiver Science Publishers B. V., North-Holland,New York 1991.Cox, J., Charlesworth, W., W., Anderson, D.c., “Domain Composition Methods ForAssociating Geometric Modeling With FiniteElement Modeling”, Proceedings of theACMISIGGRAPH Symposium on SolidModeing Foundations and CADICAMApplications, Austin, TX, June 5-7,1991.Cox, J., Ferguson, H. R. P., Kohkonen, K.,“Single Domain Methods For Modeling ObjectsIn The Round For Engineering AndManufacturing Applications”, Advances inDesign Automation, ASME Design AutomationConference, Orlando, FL, Sept. 25-28, 1988.Epstein, D.B.A., J.W.Cannon, D.F.Holt, F.V.F.Levi, M.S.Paterson, W. P.Thurston, WordProcessing in Groups, Jones and Bartlett,Boston, 1992.Farin, G. E., Curves and sur$acesfor ComputerAided Geometric Design, Academic Press Inc.,Boston, 1988.Ferguson, H., “Umbilic Torus NC,”SIGGRAPH89 Art Show, Boston, Mass.,Leonardo, Journalof the IntematiomdSociety forthe Arts, Sciences and Technology,Supplemental Issue, August 1989, page 117,122.Ferguson, H., Two Theorems, Two Sculptures,Two Posters, American Mathematical Monthly,Volume 97, Number 7, August-September 1990,pages 589-610.Ferguson, H., “Algorithms for ScientificVisualization,” Supmmmputing Research CenterTech. qort SRC-92-XXX,1992.Firby, P. and C. Gardiner, Surface Topology,John Wylie & Sons, New York, 1982.Ford, L R,, Automorphic Functions, Chelsea,New York 1951.Hoffmann, C., Geometric Modeling, MorgenKaufman,New York, 1990.Lapidus,L., Finder,G. F., Munericai Solutionsof Partial Differential Equatwns in Science andEngineering, John Wiley & Sons, New York,1982.Martin, G. E., The Foundations of Geometryand the Non-Euclidean Plane, Springer-Verlag,New York, Second Rinting, 1986.Rockwood, Alyn, “The (5,3) Toroidal Knot,”SIGGRAPH86 Art Show Catalog~ugust,1986.Siegel, C.L., Topics in Complex FunctionTheory, vol.1: “Elliptic Functions andUniformization Theory”; VO1.2,“AutomorphicFunctions and Abelian Integrals”, WylieInterscience,New York 1988

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