Abstract A method for the extrusion of arbitrary polygon meshes is introduced. This method can be applied to model a large class of complex 3-D

Transcription

1 Abstrat A method for the extrusion of arbitrary polygon meshes is introdued. This method an be applied to model a large lass of omplex 3-D losed surfaes. It onsists of defining a (typially small) set of onneted polygons in 3-D that form a skeleton of the final objet, and assigning extrusion distanes to all polygons. Both sides of a polygon may have different extrusion distanes. An automati extrusion algorithm onstruts a losed 3-D polygon mesh around the skeleton making use of the indiated extrusion distanes. We all this proess inflatingthe polygonsof the skeleton. Unlike traditional extrusion, the method works for non-planar skeleton onfigurations, and it also supports branhing skeleton strutures (i.e. edges with é2 inident polygons). 1

2 Polygon Inflation for Animated Models: a method for the extrusion of arbitrary polygon meshes C.W.A.M. van Overveld Department of Mathematis and Computing Siene Eindhoven University of Tehnology æ B.Wyvill Department of Computer Siene University of Calgary y Otober 18, 1996 keywords: geometri design, polygons, polygon mesh, omputer graphis 1. Introdution Designing 3-dimensional free-form polyhedral objets is in general an arduous task. Construtive solid geometry tehniques ([Requiha 80]) are very helpful for e.g. mahine parts or in the ontext of industrial design; they seem to lak flexibility to be very useful in appliations like harater animation or the simulation of biologial objets. Few alternatives seem to be available here: often objets are modeled by means of elaborate sampling of existing 3-D objets and subsequent modifiation (suh as the reation of syntheti ators modeled after real human ators or atresses; [Thalmann 84]). A promising alternative approah is the use of a skeleton whih onsists of few primitive geometrial objets whih an be interpreted as a distribution of (say, eletrial) harges in spae; the equi-potential surfaes at a given fixed potential value form a geometrial objet with several properties that render this so alled soft objets or impliit surfae modeling tehnique ([Wyvill 89]) well suited to harater animation or biologial simulation. Some disadvantages of impliit surfaes, however, are unwanted blending ([Guy 95]) and the bulging problem whih is still only partially 1 solved ([Bloomentha 95]); also the polygon mesh that is generated for an impliit surfae is not attahed to the loal skeleton geometry, so deforming an impliit surfae, or even moving an impliit surfae through spae auses the polygon mesh to be moved over the surfae in an unpreditable way 2, [Haumann 89], [M.P.Gasue 89], [Forsey 91] for geometri æ Address: P.O.Box 513, 5600 MB, Eindhoven, The Netherlands. wsinkvo@info.win.tue.nl y Address: 2500 University Drive N.W., Calgary, Alberta,Canada, T2N 1N4 blob@ps.ualgary.a 1 The onvolution tehnique is in most ases omputationally too expensive for pratial appliation 2 see, however [M.Desbrun 95]. Most important, the onversion from impliit surfaes to polygon meshes requires sampling, and in order to apture small shape features, high sampling densities may be neessary(see also setion 6). Although ray traing alleviates the problems related to polygonalisation of impliit surfaes, polygon mesh representations are very useful for prototyping, interative viewing and animation. Muh work has been done on parametri surfae representationson the basis of pieewise polynomial pathes,either defined by ontrol points or boundary urves ([Boehm 84], [Chiyokura 83]). Reent results allow loal and global shape modifiations based on physis based operators ([Welh 92]), and modern versions of parametri surfaes support arbitrary topologies ([Baja 92], [Loop 94]), but the emphasis of these new results seems to be on advaned surfae representation shemes rather than on the tools to reate omplex ompound surfaes from srath. In other words, few papers, if any, ould be found that address the problem of setting up the initial boundaryurve network or the initial ontrol point mesh suh that, say, a 3-D ar- 2

3 modeling tehniques that also make use of the notion of a skeleton to guide the shape of a 3-D surfae. However, these tehniques are mainly related to transferring hanges in a skeleton to the orresponding modifiations of the surfae, rather than the ab initio design of ompound 3-D surfaes. The algorithm as presented in this paper, to be alled polygon inflation, may be useful in the design stage of a 3-D polyhedral model. Sine it is in many respets similar in sope and purpose as impliit surfae modeling (ISM), It may be seen as an alternative to ISM: it also offers an intuitive shape definition via a skeleton positioned in spae, and it supports a large variety of ahievable shapes. However, it doesn t suffer from the disadvantages of bulging and unwanted blending as with impliit surfaes, and it does not rely on dense sampling for aurate representation of small shape features. It is a generalization of the familiar extrusion operation to upgrade polygons to polygonal prismati objets: it upgrades onfigurations of adjaent polygons to pieewise polygonal meshes. Both the initial polygon onfigurationandthe resultingpolygonmesh are 2-D topologialstruturesembedded in3-d spae; the resulting polygon mesh is a losed 2-manifold; the original polygon onfiguration (the skeleton) is not restrited to be one polygon, but it may be an arbitrary onfiguration of polygons instead. The mesh that results from polygon inflation may be used in onnetion with parametri surfae representations (treating the polygon mesh as a mesh of ontrol points for pieewise polynomial pathes) to allow e.g. for shape manipulation tehniques from variational modeling ([Welh 92]). Also it may be used in onnetion with an impliit surfae model (assuming the polygons from the polygonal skeleton to be also skeletal elements for an impliit surfae model that is superimposed on the resulting 3-D polyhedral model) in order to take advantage of the ollision detetion failities offered by impliit surfaes ([M.P.Gasue 91]). The requirements to deriving losed manifold 3-D models from simple pieewise polygonal skeletons an be formulated as follows: similarity: the polygonal skeleton, where eah polygon is annotated with two (possibly different) extrusion distanes is the sole speifiation of the resulting 3-D model. Therefore the 3-D model should bear a lose resemblane to the starting skeleton. flexibility: a prominent appliation of polygon inflation is omputer animation. The skeleton has few internal degrees of freedom, and hene an be animated without too muh effort by means of e.g. dynami simulation ([Overveld 94]). The resulting 3-D model would in general be too omplex to animate diretly, and therefore it is advantageous to derive its shape, for eah frame, from the moving skeleton. But this means that polygon inflation should allow for widely varying angles between adjaent polygons. smoothness: often skeleton-based modeling tehniques are used for generating (artoon-versions of) living reatures. This is in partiular the ase for impliit surfae modeling, where the smooth blends between links mimi the organi shapes of limb onnetions in nature. To be appliable in these ontexts, polygon inflation also should offer smooth blends between segments resulting from inflating several adjaent polygons from the skeleton. In partiular, for real-time purposes, rendering the resulting polygon mesh with Phong shading ([Foley 90]) should give aeptable results, without the need to replae the polygon mesh by a mesh of parametri pathes ([Boehm 84]). toon harater is shaped. Commerial geometri modeling and animation systems support funtionality to aid ontrol point plaement (extrusion (see [J.Vine 92], pp 51-52), rotational sweep, free form deformation and loal mesh refinement), but in mostasesthe animatorissupposedtomanipulatethe finalfull 3-Dmodel. Apartfrom theimpliit surfaeliterature, we have found very few results that elaborate on the proess of transforming a oarse, easy-to-model simplified objet skeleton into a smooth full 3-D model. See fburtnykwein 3

4 robustness: polygon inflation should be stable with respet to the angles between adjaent polygons. In partiular, the resulting 3-D mesh should avoid self-intersetion as muh as possible. versatility: polygon inflation should work for skeleton polygonal networks with arbitrary topology. The rest of this paper is organized as follows: in setion 2 we desribe the inflating polygon method by means of a series of inreasingly versatile approahes to the problem of extrusion of an arbitrary polygon mesh. Setion 3 is devoted to the problem of self intersetion (overlap). Arbitrary topologial onfigurations (the ase of edges with é 2 inident polygons) are studied in setion 4. Setion 5 disusses two problems with the proposed method, in partiular the limitations of the overlap orretion method. The method is ompared with impliit surfae modeling in setion 6. Finally, setion 7 ontains the summary and the onlusions. 2. Inflating polygons: free form modeling of 3-D manifolds The tehnique proposed here is based on the observation that for most people, onstruting in 3-D is essentially harder than drawing these same objets in 2-D. Nevertheless, a drawing in 2-D of a 3-D objet is essentially a projetion of that objet. The (re-)onstrution proess therefore an be seen as a (generalized 3 ) inverse projetion. In order to perform an inverse projetion, additional information is needed, apart from the projetion itself. In some subsequent steps we will onsider inreasingly sophistiated ways to furnish this information. Eah time we will use the example of the simple (more or less) human-shaped polygonal objet of figure 1a. It is drawn in side-view in figures 1b-1f. For larity, the arms have been omitted. The several steps of the algorithm are motivated by the requirements listed in setion Uniform extrusion: the entire inverse projetion is parameterised by means of one number: the distane the objet is to be translated in a fixed and given diretion (fig. 1b). This is in aordane with the similarity requirement, but it violates the flexibility requirement sine the uniform extrusion does not allow angles other than 180 degrees between adjaent polygons. 2. Non-uniform extrusion: the translation distane of every polygon is an attribute of that polygon; the translation diretion is also fixed (fig. 1), and hene flexibility is not supported. 3. Non-uniform two sided extrusion: eah polygon is extruded both in bak- and forward diretion where the two translations distanes may be hosen differently for every polygon. The translation diretion is still fixed (fig. 1d). This might support flexibility, to some small extend, but the several omponents that beome extruded perpendiularly are very likely to penetrate when bending ours. 4. C 0 non-uniform two sided extrusion: in two adjaent polygons, the translation distanes may differ. As a result, a jump (C 0 disontinuity) arises between the two extruded polygons. If, instead, the translation distane for a vertex is the average value of the extrusion distanes of the adjaent polygons, this jump vanishes and the resulting pieewise polygonal surfae beomes C 0. This applies on both sides of the extruded skeleton. See fig. 1e. This is a first attempt to introdue smoothness. 3 we will see later on why this projetion has to be alled generalized 4

5 5. Trapezoidal C 0 non-uniform two sided extrusion: The methods onsidered thus far don t perform very well with respet to the requirements of flexibility and smoothness. Indeed: sine the extrusion in both diretion has so far been a parallel extrusion, an extrusion profile very similar to fig. 1g arises. Here BE represents a side view of some original polygon; AF is the result from bakward extrusion and CD is the result from forward extrusion. Note that serious shape problems in the resulting profile our in the ase of a non-180 degree angle between two adjaent polygons of the skeleton. So in order to ure these artifats, rather than applying parallel translationsfrom BE to AF and CD, respetively, it should be saled down (shrunk) at the same time, thus produing the (double) trapezoidal extrusion profile as depited in figure 1h. The wedge-shaped area in between two adjaent trapezoids allows for spae in the ase of non-180 degree angles between the original polygons. Applying this to every polygons in isolation results in fig. 1g. 6. Trapezoidal C 0 non-uniform two sided extrusion with provisions for shared edges: The very muh segmented ( wasp-waisted ) impression of figure 1h arises due to the fat that every polygon is shrunk in isolation. The wedge-shape indents should be filled in with additional polygons. Consider the polygon onfiguration in fig. 2a. A new polygon mesh is generated as follows (this needs to be done for both extrusion diretions) æ for every original n-sided polygon, reate a shrunk n-sided polygon with all verties translated aording to the priniple of trapezoidal C 0 non-uniform two sided extrusion (a-type polygon in figure 2b). The original verties will be alled o-verties; the new ones are referred to as n-verties. æ for every edge that is shared by two adjaent polygons, reate a 4-sided polygon, onneting the 2 X 2 n-verties of the adjaent polygons (b-type polygon in figure 2b). 4 æ for every non-shared edge, reate a 4-sided polygon, onneting the 2 o-verties of the original edge and the 2 n-verties that resulted from the translation and saling (-type polygon in figure 2b). æ for every o-vertex that is ompletely surrounded by, say m, shared edges, reate an m-sided polygon whih onnets the m n-verties that arose from that o-vertex by translating and saling (d-type polygon in figure 2b). æ for every o-vertex that is not ompletely surrounded by shared edges, but that has at least one inident shared edge, reate a triangle for every one of its surrounding shared edges. The apex of this triangle is the o-vertex proper, the two base verties are two n-verties that arose from that o-vertex by translating and saling (e-type triangle in figure 2b). æ for every o-vertex that has no inident shared edges (edges, shared with an adjaent polygon of the skeleton), no further adjustments have to be made. Applyingthe above steps, results in the polygonmesh of fig. 2b. Notiethat the intermediate polygons, that were added to ope with the onnetions of adjaent segments over the shared edges, signifiantly ontributes to the smoothness of the resulting polygon mesh. This is similar to the idea of orner utting, used in the generation of reursively refined polygonal meshes to obtain smoothly urved surfaes ([Boehm 84], [Doo 78]). Observe that this algorithm assumes that in the initial polygon mesh all edges belong to at most 2 polygons. Moreover, extreme non-onvex polygons may produe ounter-intuitive results in the 4 This polygon may not be planar, and splitting it into 2 triangles may therefore be neessary. 5

6 sense that the polygon that is onstruted for an a-type polygon p may projet partially outside p. The next setion addresses the problem of versatility (i.e. generalizing inflated polygons to onfigurations with shared edges that have more than two inident polygons). So far, the design strategy as proposed here is able to generate quite involved and intuitive shapes starting from a planar polygon mesh and only two numerial attributes per polygon. In one final extension we release the onstraint of the input polygon mesh to be planar, to improve the robustness of the algorithm Trapezoidal C 0 non-uniform two sided extrusion with provisions for shared edges and non-planar input meshes: Suppose that the input mesh takes part in a 3-D animation sequene. For most types of more or less realisti 3-D motions, the requirement of planarity of the input mesh is unaeptable. In steps 3-6, the translation vetor for the extrusion was the normal vetor of the polygon to be extruded. However, for a non-planar polygon, this normal vetor is not defined. Instead, we use the following strategy. For a vertex we take the normalized ross-produt of the inident edges as the diretion of the normal vetor in that vertex. If a polygon happens to be planar, all these normal vetor diretions are parallel; otherwise, the normal vetors vary from vertex to vertex, but in every vertex the normal vetor is perpendiular to the region of the warped polygon lose to that vertex. It an be seen that the above onstrution leads to a losed 2-manifold polygonal mesh where every edge is shared by preisely 2 polygons. 3. Correting self intersetion With our method, self intersetion annot be avoided globally. However, on a loal sale, we an avoid self intersetions for the extruded parts of two adjaent polygons by defining for eah polygon that is to be extruded a safe region in 3-D spae, suh that its extruded part stays in this safe region. If we assure that the safe regions of two adjaent polygons are disjoint, the extruded potions of these polygons annot self interset. A safe region is bordered by separator planes. For two adjaent polygons p 1 and p 2, sharing an edge e, their ommon separator plane s passes through e. As long as the extruded version of p 1 stays on p 1 s side of s and the extruded part of p 2 stays on p 2 s side of s, these two extruded parts annot get into eah other s way. Notie that it doesn t matter what the diretion of s is as long as it passes through e; however, for the sake of symmetry it is best to make s the bisetor plane of p 1 and p 2. To study the problem of self intersetion somewhat loser, and see how the safe regions-idea works out, first onsider the 2-D onfiguration, in fig. 3a with a set of line segments that are to be extruded AB, BC, CD. The overlap areas learly show up. The safe regions that are needed to keep the extruded parts away from eah other are defined by introduing the two separators: A, B, C, through B, in the diretion of the bisetor of AB, BC, andb, C, D, through C, in the diretion of the bisetor of BC, CD. Point P has to be shifted parallel to BA to the left to move it to the lower left of separator A, B, C, andq has to be shifted up parallel to BC to move it to the upper right of the same separator. This re-orients PQto arrive at the orreted P 0 Q 0. Similarly, R has to be shifted down parallel to CB to move it to the lower left of separator B, C, D and S has to be moved right parallel to CD to move it to the upper right of the same separator. This re-orients RS to arrive at the orreted R 0 S 0. In figure 3b the overlaps have been orreted. 5 By the input polygon mesh we mean the skeleton that is to be inflated; the output mesh is the result of the inflation proedure 6

7 In the 3-D ase, AB, BC et. represent polygons, say p 1 and p 2. The separator beomes a plane whih passes through the shared edge e of two suh adjaent polygons. The orientation of the normal of a separator plane is the average of the two normalized normal vetors, with appropriate mutual orientations, of the polygons p 1 and p 2. The extruded points of polygons p 1 are fored to stay on p 1 s side of the separator plane, and the extruded points of polygon p 2 are fored to stay on p 2 s side of the separator plane. In order to arrive at the new positions of a orreted point the onstrution from figure 3 is applied. Let v be a point whih resulted from extrusion from polygon p 1. First it is projeted perpendiularly onto p 1 to yield v p.nextv p is projeted perpendiularly onto e to yield v pp. The diretion v p v pp defines a diretion to projet v onto the separator plane. This yields the point w. Finally, a point x on a fixed distane (say, half of jv p, v pp j) from w is defined, whih lies on the safe side of the separator. This point will replae v if, originally, v and v p were on different sides of the separator plane (so the point v 0, whih is on the same side as v p, is not replaed by x 0 in figure 3) 6. The above onstrution serves to keep extruded parts from overlapping of the inflated polygon mesh at the onave sides of shared edges. A seond, albeit less onspiuous, form of overlap may our at the onvex side. This is indiated in figure 3d. The remedy here again onsists of shifting the extruded points, but instead of the separator plane to indiate the safe region, we have to use a plane whih is perpendiular to the separator plane. The geometrial onstrution is very similar to the previous ase, and is not further detailed here. 4. Dealing with arbitrary topologial onfigurations. The onstrution of the resulting polygon mesh in the ase of a 2-manifold (that is, all edges have at most 2 inident polygons) is relatively straightforward. For an edge e with é2 inident edges, a slightly more elaborate approah has to be followed. The proess onsists of 4 phases. See figure Projet all inident faes on a plane perpendiular to e. Designate one of the faes as a referene fae, and sort the faes in the order of inreasing ounter lokwise angle with the referene fae. Orient the normal vetors in a ounter lokwise diretion. The diretion of the normal vetor will be alled the UP -side of a fae; the other side is the DOWN -side. 2. Generate the offset fae for all inident faes. For eah extreme vertex v of e, reate a list with the orner verties of the offset faes near v. 3. Generate the b- and -type 4-sided polygons as defined in setion 2 under point 6. The b-type polygons assoiated with e are formed by onneting the UP -side offset fae for inident fae i with the DOWN -side offset fae for inident fae i +1and loop over all inident faes. The edges of these quadrilaterals that don t border an a-type polygon are also stored in a list that is kept with vertex v. These edges will be alled the vertex-related edges of the b- and -type polygons, as opposed to the edge-related edges that are shared with an a-type polygon. At the end of this pass, every vertex v of the original mesh will have two lists: one list of verties, alled l v,i.e.the orner verties of the a-polygons losest to v, and a list of edges, l e, i.e. the vertex-related edges of the b- and -type polygons losest to v. 4. For every vertex v, try to onstrut losed loops of the edges in v:l e. In one suh loop, two subsequent edges share one vertex of v:l v. Notie that it is possible that there are edges in v:l e that do not our in a loop; it may even happen that no losed loop an be formed. In partiular, if v 6 This onstrution avoids overlapping extruded polygons only for adjaent polygons; overlaps between extruded polygons that result from non-adjaent original polygons are not deteted. Moreover, in the ase of arbitrary large extrusion distane, the point x may end up on the wrong side of one of the other separator planes assoiated to the original polygon. This is not heked either. 7

8 has adjaent -type polygons, the list v:l e will ontain isolated edges (the orner-related edges of these -type polygons). For eah isolated edge e we form a triangle with the point v and the two extremes of e as orners. For eah losed loop found, say with n edges, we form n triangles that have a ommon vertex namely v, and subsequent edges from that loop as their opposite sides. 5. Problems. The polygon inflation tehnique turns out to be quite versatile and intuitive in using. Still, there are two ases that ould be problematial. The self intersetion orretion works only loally between adjaent polygons. A part of the mesh that results from polygons further along the skeleton ould still overlap with a loally orreted part of the mesh. This problem ommonly ours with the usage of surfae models in animation. A seond problem ours when applying the overlap orretion, and a point is translated to one side of a separator plane. It an our that the point has to be translated so muh that it is positioned on the wrong side of one of the other separator planes. This problem either ours when a very large thikening width is applied to a rather small polygon and/or two adjaent polygons meet at a very aute angle. 6. Comparison with impliit surfae modeling The main motivation for developing the inflating polygons tehnique is to have a omputationally less omplex alternative for impliit surfae modeling. In the table below we summarize the main features of the inflating polygons tehnique in omparison with impliit surfae modeling. A quantitative omparisonismadeinsetion7. 8

9 feature impliit surfaes modeling inflating polygons definition skeleton with a variety of skeleton is a polygonal mesh of types of primitives generated surfae smooth 2-manifold polygonal 2-manifold integrity has inside and outside requires some are to avoid self intersetion blending yes no suffer from yes, although no unwanted bulging partial solution exists([bloomentha 95] support for ollision yes no detetion visualizing requires post proessing heap: result is a polygon (tessellation or ray traing) mesh smoothness naturally smooth requires polygonal post proessing (e.g orner utting) ompatness 7 tessellation method determines for a given number of input ompatness; number of polygons, a fixed number of triangles generated depends output polygons is reated on the size of the surfaes and required sampling auray for small shape features arbitrary yes yes topology For a quantitative omparison, we have modeled the same objet both with impliit surfaes and with polygon inflation. We start with a simple polygonal skeleton that is transformed into a 3-D objet by means of polygon inflation followed by one pass of subdivision. This gives rise to 218 polygons, omputed in about 0.5 seonds. Next we have used the same polygon mesh as skeletal elements for impliit surfae modeling with a oarse sampling density, and similarly with a twie as high sample density. The number of resulting polygons was 1500 and 6372, respetively omputed in approximately 8 and 30 seonds. So we observe that in order to get a shape representation with polygonalised impliit surfaes that is omparably aurate as with polygon inflation, we need onsiderably more polygons, and hene muh more omputational effort. See the olor plate setion for rendered images of these experiments. 7. Summary and onlusion The input for the polygon inflation algorithm is an arbitrary polygon mesh. In every vertex, a surfae normal vetor has been obtained by means of normal vetor averaging. In every polygon, two numerial attributes are given, one for the extrusion distane in the forward diretion and one for the extrusion in the bakward diretion. It is assumed that the normal vetor assignment is onsistent, i.e. all normal vetors point towards the same side of the surfae. 9

10 The output, whih is produed automatially, onsists of a polygon mesh whih onsists either of two disjoint losed manifolds (one inside the other) in the ase the input objet was a losed manifold (e.g. a ube) or of one losed manifold (in all other aeptable ases). The resulting objet depends on the input objet in a ontinuous and intuitivemanner, whih renders the method very well suited for appliation in an animation ontext: the motion speifiation takes plae using the input mesh, and polygon inflation takes plae for every generated frame. The omputational omplexity of polygon inflation is proportional to the number of generated polygons, and is typially of the order of a few hundred or less. This means that a turn-around yle (interative modifiation of the skeleton and the omputation -from srath- of the modified resulting mesh) is of the order of one or two seonds. The omputation of a suffiiently aurate mesh for an impliit surfae model usually takes about ten times as long. Still, a faster response, useful for interative modeling, an be ahieved as long as the topologial struture of the skeleton does not hange sine for every point of the resulting mesh, it is known how it depends on verties of the input skeleton. So interative manipulation of input skeleton verties, as well as hanges to the extrusion distanes of the skeletal polygons an be translated immediately to new positions of the affeted verties of the resulting mesh 8. Topologial hanges, i.e. the reation or deletion of skeletal polygons or hanges of the onnetion topology is more involved and requires re-alulation of the mesh topology. In olor plates A, B, C and D the working of the algorithm is shown. In olor plate A (upper left) a flat shaded polygonal model is shown of a four-legged animal (whih is suffiiently simple for real time interative motion speifiation and motion-play-bak). The upper right image shows the result from polygon inflation. A post proessing step in the form of relaxation ([Mallet 92], [vo95]) has been applied to arrive at a somewhat smoother mesh (lower left). The lower right image depits the same model rendered with smooth shading. In olor plate B the same sequene is repeated for a duk, exept that the method used for smoothing (lower left) onsists of a polygon subdivision step. Color plate C shows an objet that was designed to show the apability of the algorithm to deal with arbitrary topologies: it ontains a hole and an edge with 3 inident polygons. It was proessed and rendered in a similar way to the duk sequene. Finally, in olor plate D we give the results of the quantitativeomparison between impliit surfae modeling and polygon inflation. Upper left: the polygonal skeleton used in both models. Upper right: the result from polygon inflation followed by one pass of subdivision. Lower left: oarsely sampled impliit surfae (1500 triangles) whih has onspiuous sampling artifats; lower right: impliit surfae sampled with a twie as high sample density (6372 triangles). Notie that with this higher sample density, the smooth inner region of the surfae is approximated well, but the thin points still show sampling artifats. On the other hand, the inflated objet only onsists of 218 polygons. Aknowledgment This work is the result of a 6-months ollaboration of the two authors while one of them was on a sabbatial leave in Calgary. This author (CWAMvO) therefore wishes to thank the Department Board of the Dept. of Mathematis and Computing Siene of Eindhoven University of Tehnology to give him the opportunity of his leave, as well as the Department of Computer Siene of the University of Calgary. This work was partially supported by the Natural Siene and Engineering Researh Counil of Canada. 8 in our implementation, we do not support this interative editing. Instead, our skeleton-inflation program is implemented as a UNIX filter. 10

11 a b d e f figure 1. a: original figure; front. b f: profile b:uniform extrusion; :non uniform extrusion; 11 d:non uniform 2 sided extrusion; e:c 0non uniform 2 sided extrusion; f:non uniform 2 sided extrusion with trapezoidal extrusion profile.

12 o type vertex n type vertex a,b,,d indiate polygon types; all non labeled triangles are of type e. a b a b a b a b d b a b a Figure 2a (top): a polygon mesh before applying subdivision; 2b (bottom): the result of the subdivision algorithm. 12

13 C R D A F S P E B A B Q C D Figure 1g: parallel extrusion gives overlap problems for non-180 degree angles between adjaent polygons. Figure 3a: the original onfiguration is indiated by the thik lines AB, BC, CD. The thikened area is shaded, and the overlapping regions show up white. C D S P Q R separator B-C-D A B separator A-B-C Figure 1h: trapezoidal extrusion allows adjaent polygons to have non-180 degree angles. Figure 3b: overlap orretion has been applied. The points P...S have been shifted to their new positions P...S at the safe sides of the separators. The shifting diretions have been indiated by dashed lines; these are parallel to AB, BC and CD, respetively. 13

14 Figure 4. The four phases of the mesh omputation for a non-manifold edge. Upper left: the inident faes are sorted on inreasing angle with a referene fae, and normal vetors are oriented aordingly. Upper right: for eah inident fae, two offset faes are omputed Lower left: offset fae UP for inident fae i is onneted with offset fae DOWN for inident fae i+1, and yli. Lower right: the remaining triangular holes are filled. Inset: as a onsequene, the marked point will our in the resulting mesh. 14

15 Referenes [Baja 92] C. Baja and Insung Ihm. Smoothing Polyhedra using Impliit Algebrai Splines. Computer Graphis (Pro. SIGGRAPH 92), 26(2):79 88, [Bloomentha 95] Jules Bloomenthal. Bulge Elimination in Impliit Surfae Blends. pages 7 20, [Boehm 84] W. Boehm, G. Farin, and J. Kahman. A survey of Curve and Surfae methods in CAGD. Computer Aided Geometri Design, 1:1 60, [Chiyokura 83] [Doo 78] H. Chiyokura and F. Kimura. Design of solids with free-form surfaes. Computer Graphis (Pro. SIGGRAPH 83), 17(2): , D. Doo. A subdivision algorithm for smoothing down irregular shaped polyhedrons. In Pro. Conf. Interative Tehniques in CAD, Bologna, Italy, pages IEEE, [Foley 90] James D. Foley, Andries van Dam, Steven Feiner, and John Hughes. Computer Graphis Priniiples and Pratie. Addison-Wesley, [Forsey 91] David R. Forsey. A surfae model for skeleton-based animation. pages 55 74, [Guy 95] Andrew Guy and Brian Wyvill. Controlled Blending for Impliit Surfaes using a Graph. pages , [Haumann 89] John Chadwik David Haumann and Rihard Parent. Layered Constrution for Deformable Animated Charaters. Computer Graphis (Pro. SIGGRAPH 89), 23(3): , August [J.Vine 92] J.Vine. 3-D omputer animation. AddisonWesley, [Loop 94] [Mallet 92] Charles Loop. Smooth Spline Surfaes over Irregular Meshes. Computer Graphis (Pro. SIGGRAPH 94), 28: , July J.L. Mallet. Disrete smooth interpolation in geometri modelling. Computer Aided Design, 24(4): , April [M.Desbrun 95] M.Desbrun, N. Tsingos, and M.P.Gasuel. Adaptive sampling of impliit surfaes for interative modeling and animation. pages , [M.P.Gasue 89] M.P.Gasuel. Welding and Pinhing Spline Surfaes: New Methods for Interative Creation of omplex objets and Automati Fleshing of Skeletons [M.P.Gasue 91] M.P.Gasuel, Anne Verroust, and Claude Pueh. Animation and Collisions between Complex Deformable Bodies [Overveld 94] [Requiha 80] C.W.A.M van Overveld and Hyeongseok Ko. Small steps for mankind: toward a kinematially driven dynami simulation for urved path walking. The Journal of Visualisation and Computer Animation, 5: , A.A.G. Requiha. Representations for Rigid Solids: Theory, Methods, and Systems. ACM omputing surveys, 12(4): , Deember

16 [Thalmann 84] D. Thalmann and N. Thalmann. Animated types and ator types in omputer simulation and animation. Pro. SCS Conf. on Simulation in Strongly Typed Languages. San Diego, pages 51 56, [vo95] C.W.A.M. van Overveld. Pondering on disrete smooth interpolation. Computer Aided Design, 27(5): , [Welh 92] [Wyvill 89] William Welh and Andrew Witkin. Variational surfae modeling. Computer Graphis (Pro. SIGGRAPH 92), 26(2): , July Brian Wyvill and Geoff Wyvill. Using Soft Objets in Computer Generated Charater Animation. Computers in Art, Design and Animation, pages ,