A Discrete Spring Model to Generate Fair Curves and Surfaces

Transcription

1 A Discrete Spring Model to Generate Fair Curves and Surfaces Atsushi Yaada Kenji Shiada 2 Tootake Furuhata and Ko-Hsiu Hou 2 Tokyo Research Laboratory IBM Japan Ltd. LAB-S Shiotsurua Yaato Kanagawa JAPAN ayaada@jp.ib.co furuhata@jp.ib.co 2 Mechanical Engineering Carnegie Mellon University ttsburgh PA 523 shiada@cu.edu khou+@andrew.cu.edu If the paper is accepted one of the authors will present the paper at the Pacific Graphics 99 conference. Abstract To generate fair curves and surfaces is an iportant tool in the area of coputer graphics (CG) coputer-aided design (CAD) and other geoetric odeling applications. In this paper we present an iteration-based algorith to generate fair polygonal curves and surfaces that is based on a new discrete spring odel. In the spring odel a linear spring which length approxiately represents a curvature is attached along the noral line of each polygon node. Energy is assigned to the difference of the lengths that is difference in curvature of nearby springs. Our algorith then iniizes total energy by iterative approach. Our algorith accepts as inputs () an initial polygonal curve (surface) which consists of a set of polygonal segents (faces) and a set of nodes as polygon-vertices and (2) constraints for controlling the shape. The outputs are polygonal curve (surface) in sooth shape. We also describe a ethod for iproving perforance of our iterative process into a linear execution tie. Our algorith provides a tool for fair curve and surface design in interactive environent. Keywords: geoetric odeling fair surface design polygonal odels energy iniization. Introduction The purpose of this paper is to provide an algorith to generate fair curves and surfaces used in the fields of coputer graphics (CG) and coputer-aided design (CAD). Generation of fair shapes is a ajor topic in shape design [5][0][7][9][22][23][24][26]. It is also required in other applications such as sooth shape fitting to scattered points [7][20] texture apping [5] and so on. Curves and surfaces treated in this paper are represented in polygonal for. The inputs of the algorith are () an initial polygonal curve (surface) consisting of a set of nodes and a set of polygonal segents (faces) and (2) constraints for controlling the shape. The algorith oves the nodes to suitable positions iniizing curvature variation under the given constraints. The outputs are polygonal curve (surface) in sooth shape. The polygonal surfaces treated in this paper are not liited to triangular eshes. Quadrilateral eshes are also available and theoretically n-sided faces such as pentagon or hexagon ay also be included. Our algorith is classified into iterative approach of Gauss-Seidel type: node positions are iteratively updated under the given constraints. To update the node positions two types of spring forces are applied to each node: () a force acting in the noral direction to optiize the curvature variation and (2) a force in the vertical direction of the noral to optiize node distribution. Main idea of this paper lies in the discrete spring odel producing the forer force. The spring odel works to iniize curvature variation. Curvature is a natural easureent of fairness; therefore our spring odel produces fair curves and surfaces. As Taubin [23] points out ost energy-iniization approaches are expensive in ters of tie and space. This paper also provides a ethod for achieving a linear execution tie. The reainder of the paper is organized as follows. In Section 2 we suarize previous work. After defining a discrete spring odel in Section 3 we present a curve (surface) odeling algorith based on the spring odel in Section 4. Section 5 show soe results obtained by using the algorith and Section 6 suarizes the paper. 2. Previous Work A considerable aount of work has been done on fair surface odeling. We consider that ost previous work related to polygonal surface odeling can be classified into two types: finite-difference approaches and finite-eleent approaches. Our approach belongs to the forer category. In both types of approaches node positions are updated iteratively or are calculated by solving a large sparse linear syste. In the finite-difference approaches values are evaluated only at nodes while in the finite-eleent approaches values are evaluated over faces by using nuerical integration. One key-factor to distinguish several approaches in a category is the spring odel they eploy to ove the node positions. Irrespective of the classifying types when the displaceent of deforation is large that is when the shape of the initial polygonal surface is very different fro the final shape the deforation reduces to the non-linear proble. Therefore once solving a sparse linear syste does not yield a suitable answer; the syste ust be solved iteratively. The cases of large deforation appear often in the process of surface design. To construct a robust algorith for such cases we select iterative approach. Two following subsections survey both types of approaches targeting surface odeling. In the third subsection soe other applications that use the fairing as a part of their algorith are also surveyed. 2. Finite-Difference Approaches for Surface Modeling Laplacian soothing is the easiest way to generate a fair surface. This approach iteratively oves the position of a node to the barycenter of its neighboring nodes. It has the characteristic that the area of the surface is

2 iniized under given constraints. This approach is often used to iprove the geoetrical irregularity of a esh in the field of finite-eleent eshing []. But when this approach is applied to surface odeling the proble arises that a sharp tip is generated in the neighborhood of a node fixed by a constraint. Moreover it is not possible to control norals by giving noral constraints. For surface odeling Szeliski and Tonnesen [22] used a particle syste in which each particle is defined by its position and noral and the population of particles is controlled autoatically. To obtain a fairly triangulated surface they iniized the weighted su of three energy factors: () co-planarity (2) co-norality and (3) co-circularity. The co-planarity condition causes neighboring nodes to lie on each other's tangent plane the co-norality condition odifies irregular twisting of two neighboring nodes and the co-circularity condition preserves constant curvature on an edge connecting neighboring nodes. In coparison with their approach one factor by our spring odel provides a siilar effect to their three factors. Mallet [6][7] provided a general forulation for discrete surface interpolation with any adjustable paraeters. When haronic weighting [7] is selected aong his proposed paraeters - as in his results - his approach iniizes the su of the distances each of which is fro a node to the barycenter of its neighboring nodes. Taubin [23] proposed a fair surface design approach in which high-frequency ters such as noise are reoved by using the technique based on Fourier analysis. Due to its linear execution tie his approach is powerful for polygonal surfaces with illions of nodes for exaple one captured by using a range scanner. A node is iteratively oved to a suitable position on a line connecting a node and the barycenter of its neighboring nodes. The suitable positioning is deterined so as to avoid shrinkage of the entire shape. Basically Taubin s approach is suitable for reoving noise fro a given initial shape without causing shrinkage. The ain difference between our approach and those of Mallet and Taubin is that in our approach curvature variation is used as a easureent of iniization. Curvature is a natural easureent of fairness; consequently our approach produces a fairer shape than theirs. Welch and Witkin [26] also proposed a esh-based odeling ethod for surfaces with arbitrary topology. After defining a local surface equation in the neighborhood of each node which fits the latter s neighboring nodes in the least-square sense they iniized a fairness nor based on curvature which is calculated fro the local surface. The local surface is teporarily used to evaluate the curvature at a node and the final outputs are only node positions not surfaces of faces. In the sense that curvature is adopted as a fairing easureent our approach is siilar to theirs. However their approach has to solve a 5 least-square linear syste (where is the nuber of neighboring nodes) for each node at each iteration. This is a tie-consuing process if the nuber of nodes is large. 2.2 Finite-Eleent Approaches for Surface Modeling In coparison with finite-difference approaches finite-eleent approaches generally yield a higher-quality surface and generate an explicit surface equation of each face while their coputational tie is ore expensive. Celniker and Gossard's approach [5] generates a C continuous surface by using a finite-eleent technique. They applied Zienkiewicz's shape function [27] which was originally proposed for the finite eleent analysis to attain C continuity in surface odeling. Their approach iniizes a weighted cobination of the energy factors of a ebrane and a thin plate. Welch and Witkin [25] also proposed an approach very siilar to that of Celniker and Gossard although the forer has ore general forulations for both energy factors and shape control constraints. Moreton and Sequin [9] presented a different type of finite-eleent approach in which they use biquintic Bezier patches and a fairness nor based on easures of curvature variation. Their approach produces the ost ipressive surface but it is too expensive for use in an interactive environent. 2.3 Other Applications Using Fairing Halstead and his colleagues [0] provided a solution to the proble that Catull-Clark's subdivision surface schee [4] gives a shrunken surface without interpolating given control points. In their process a fairness factor proposed by Celniker and Gossard is used in conjunction with the Catull-Clark schee. The target of Eck and Hoppe's work [7] is to generate tensor product surfaces fro scattered points. They use a thin plate as a fairness factor to reove undulations in surfaces. Levy and Mallet [5] applied their discrete sooth interpolation approach [6][7] to the non-distorted texture-apping proble. Koch and his colleagues [4] applied the thin-plate approach to a syste for siulating facial surgery and DeCarlo and his colleagues [6] applied the thin-plate approach to geoetric odeling of huan faces. As can be seen in the above literature in this subsection fairness factors are widely used in the area of CG and CAD alongside factors unique to each study which depend on their applications. Our approach has possibility to be applied for such applications. 3. Definition of the Discrete Spring Model: V-Spring Figure shows a planar curve and its noral lines at soe sapling points on the curve. Consider the noral lines at two neighboring sapling points and. For a curvature continuous curve if approaches the intersection point H of the noral lines converges to a center of curvature at [3]. Therefore our idea is to attach a linear spring as shown in Figure 2(A) to each noral line of a node consisting in a polygonal curve (surface). The linear spring works to keep equal the spring lengths - H and - H of a V-shape fored by H and. The spring length approxiately represents a curvature; therefore the action to keep equal the spring length is equal to iniize variation in curvature. Suppose node be fixed by a constraint. If length - H is saller than - H as shown in Figure 2(A) node oves to a new position along the noral in the direction to enlarge length - H to the size of - H. In contract if - H is larger node oves to a new position along the noral in the direction to shorten length - H to the size of - H. In a stable configuration and are considered to be on a circular arc whose center is at H and whose curvature radius is - H (= + d - H ). Because of this V-shaped configuration of virtual springs our spring odel is naed "V-Spring." 2

3 Nj because of issing of H. However Equation (5) does not use H in direct; therefore our spring odel is stable even for the case of parallel noral lines. In case of a planar curve noral lines along and Nj always have an intersection. However if we consider a non-planar curve or a surface noral lines do not always intersect at a point. Therefore we can not use Equation () as they are for the non-planar cases. Instead of an intersection point H in Equation () we use Hi and Hj which are feet of the shortest line segent connecting two noral lines (see Figure 2(B)). Then for non-planar cases we odify Equation () to the following equations: H Hi = ti (6) Hj = tj Nj. Figure : Planar curve and noral lines on soe sapling points. Here we forally define a displaceent d fro a current position to a new position of a node by the force of the spring odel under the supposition that is fixed. Let and be two nodes and let and Nj be unit noral vectors associated with the nodes respectively. We assue an inner product ( Nj) to be positive; therefore if two noral vectors with their negative inner product are given let either or Nj be the direction reversed vector. An intersection of the two noral lines is denoted by H (see Figure 2(A)). Let ti and tj be real values satisfying: H = ti () H = tj Nj. ti and tj correspond to the distances - H and - H respectively because and Nj are unit vectors. We define the displaceent d of node by our spring odel as: ( tj ti). d = (2) (tj - ti) in the equation is calculated as follows. For the equation that H is deleted fro Equation () by calculating inner products with and Nj following equations are obtained. (( j + ti tjnj) ( ) ) = 0 (( + ti tjnj) ( Nj) ) = 0. P (3) By solving Equation (3) in ters of ti and tj: ti = tj = (( P j ) ( + cos( a) Nj) ) 2 ( cos ( a) ) (( ) ( cos( a) + Nj) ) 2 ( cos ( a) ) where cos(a) = ( Nj ). Consequently (tj - ti) in Equation (2) is deterined as: (( j ) ( + Nj) ) ( cos( a) ). tj ti = P + (4) Therefore Equation (2) is written as: (( ) ( + Nj) ) + ( Nj) d =. (5) The denoinator of Equation (4) is always non-zero value because cos(a) (=( Nj)) is assued to be positive. One ay consider fro Figure 2(A) that nuerical error happens when two noral lines are parallel For non-planar cases the equations corresponding to Equation (3) are: (( j Hi) ( ) ) = 0 (( Hj Hi) ( Nj) ) = 0. H (7) By solving Equation (7) (tj - ti) in Equation (2) is derived as the sae equation as Equation (4). Nj Nj Hj H Hi (A) (B) Figure 2: Definition of the spring odel. (A) Displaceent d for a planar curve. (B) Displaceent d for a non-planar curve or a surface. d d 4. Curve/Surface Modeling Using a Discrete Spring Model 4. Overview of the Algorith A polygonal surface is defined as a pair of a set of nodes (i =... n) and a set of polygonal faces. We define neighboring nodes (j = 2 ) of to be a set of nodes connected to by polygonal edges. Our algorith can be classified as an iterative ethod of Gauss-Seidel type. The positions and norals of nodes are updated in each iteration and the iterations are continued until the terination condition is satisfied. Figure 3 shows an overview of our algorith. 3

4 Let be the position of the i-th node Let n be the nuber of nodes While( terination condition is not satisfied ){ For( all nodes (i =... n) ){ Step : Calculate pseudo-noral Step 2: Calculate displaceent d caused by the force exerted by V-Spring Step 3: Calculate displaceent u i caused by the force for regularizing node distribution Step 4: = + ( d + u i ) Figure 3: Overview of the algorith. The following subsections and 4.4 describe Steps 2 and 3 in Figure 3 respectively. In Section 4.5 we describe several constraints. After describing the terination condition in Section 4.6 we give a ethod for reducing an execution tie in Section 4.7. Up to Section 4.7 our discussion concerns surface odeling. Section 4.8 extends the discussion to curve odeling. 4.2 Pseudo-noral Calculation The first step of the algorith is to calculate a unit noral vector for each node of a polygonal surface (Step in Figure 3). The polygonal surface is a discrete odel; therefore the unit noral vector ust be calculated only approxiately. In our ipleentation we calculate the unit noral by averaging the norals of polygonal faces adjoining to the node. There are ore sophisticated ways to calculate the noral. For exaple one way is to calculate the noral fro a sphere fitted to the target node and its neighboring nodes in the least-square sense. However the way is ore tie consuing and fails when the target node and its neighboring nodes are on the sae plane or the neighboring nodes are placed to have the target node a saddle point. Another way used by Taubin [23] is calculate the noral as the average of vectors each of which is fro target node to one of neighboring nodes. It is the faster way; however it fails when the target node and its neighboring nodes are on the sae plane. Our choice is the ore basic but the ore robust way. In the early phase of iterations the noral vector is unreliable; however as the iterations proceed the noral vector converges in the reliable noral of the fair surface. 4.3 Node Displaceent by the Force of V-Spring Node obtains forces fro its neighboring nodes (j =... ). Each force works to keep the edge fro to in a circular arc. Weighted average of the forces akes the node ove to a new position along noral by the displaceent d (see Figure 4(B)). The way to calculate the displaceent dj of node by the force of one neighboring node is as follows. Let and Nj be unit noral vectors of nodes and respectively and let Hi and Hj be feet of the shortest line segent connecting two noral lines along and Nj respectively. Fro Equation (2) and (4) the displaceent dj is calculated as follows: ij ( tj ti) = (8) tj ti = (( ) ( + Nj) ) + ( Nj) In the sae anner we calculate d... d for neighboring nodes. The final displaceent d of node is deterined by weighted-averaging of the displaceents dj (j =... ) as follows:. (9) i = wj dj wj (0) j= j= where wj (j =... ) are weights for averaging. In our ipleentation the weight wj is deterined by the inverse of the length of the edge connecting and. wj = P i ( j = 2... ). d - + (A) Figure 4: Node displaceent by forces exerted by neighboring nodes. (A) Planar curve case. (B) Surface case. 4.4 Node Displaceent for Regularizing Node Distribution In finite-difference approaches it is iportant to aintain the regular node distribution during the iteration process because uneven distribution of nodes results in incorrect estiation of curvature. To obtain the regular node distribution we use a variation of Laplacian soothing operator [] which is a popular and effective way to reove the irregularity. The Laplacian operator oves a node to the barycenter of its neighboring nodes. However applying the regular Laplacian operator offsets the displaceent d in Equation (8). Therefore our idea is to use only a coponent ui being vertical to the noral of the displaceent of the Laplacian operator. Only using the coponent akes two displaceents d and ui being vertical each other. Therefore the two displaceents do not offset each other. The displaceent ui is written as follows: u i u0 i = = u0 i j = 4.5 Constraints u i u i d = P2 (B) P P ( u0 i ). Constraints are considered to be external forces for controlling shape of a surface. Various kinds of constraints can be considered depending on the requireents fro applications. In this section we describe several 4

5 iportant constraints used in surface odeling. We classify the constraints into two types: direct and indirect constraints. Direct Constraints Direct constraints are given directly for a certain node. Major constraints are: - Positional Constraint - Noral Constraint. The positional constraint fixes a node to a certain position during the iterations; therefore Step 2 and 3 in Figure 3 is skipped for the fixed node. The noral constraint fixes a noral of a node to a certain direction. For the noral fixed node the pseudo-noral calculation (Step in Figure 3) is skipped and the given noral is assigned. In our approach positional constraints ust be given at least to end nodes for the case of an open polygonal curve and to boundary nodes for the case of an open polygonal surface. That is why usage of Laplacian operator causes the shrinkage of the shape. If it is required to odify the shape of boundary curves of a surface start fro the odeling of boundary curves and go on to the odeling of the surface bounded by the boundary curves. Indirect Constraints Indirect constraints do not have a direct connection to a certain node. During the iterations connection between the constraints and the nodes is updated dynaically. One ajor constraint we introduce here is scattered points. It is an iportant application in CAD and CG to generate a sooth surface fitted to scattered points in least-square sense. In the application the constraint by a scattered point works to its closest point on a surface. Therefore in our discrete odel we update the connection between the scattered point and its closest node during the iterations. We have to odify slightly the algorith to introduce the indirect constraints; the odified algorith is shown in Figure 5. In Figure 5 all the steps except Step 2 and 5 are the sae as in Figure 3. In Step 2 each scattered point is connected to its nearest node. In practice it is not necessary to perfor Step 2 at every iteration; once in every several iterations is enough. To find the nearest node it is enough to search neighboring nodes in first and second orders of the previous node except when perforing a first search. In Step 5 external forces fro connected scattered points Vj (j =... ) are applied to a node. The displaceent of by Vj is calculated as: c ij = kj ( Vj ) () where kj denotes a weight assigned to Vj. The displaceent cij is interpreted as a coponent of vector Vj- along the noral. In the sae anner we calculate ci... ci for all connected scattered points. The final displaceent of node by forces of its connected scattered points is deterined by weighted-averaging c ij (j =... ) as follows: c i = wj c ij j= j= wj where wj (j =... ) are weights for averaging. In our ipleentation the weight wj is deterined by the inverse of the distance fro node to its foot Qj to the tangent plane at (see Figure 6(B)). wj = P i Qj ( j = 2... ). 5 When Qj is identical or very close to sufficiently large value is assigned to its wj. In Equation () weight kj is used to take the trade-off between fairing and keeping proxiity to the scattered point Vj. Larger kj approxiates Vj closer. In the case of a planar curve least-square fitting is geoetrically interpreted that when a spring is attached to each line fro a scattered point to the nearest point on a curve the su of the internal energies of the springs are iniized (see Figure 6(A)). Our approach is geoetrically interpreted as being to attach a spring to a line fro a scattered point to its foot in the tangent plane at the nearest node (see Figure 6(B)). We therefore consider that our approach is approxiately equivalent to least-square fitting. The advantage of our approach is that it can be applied to not only surfaces with regular topology such as tensor product surfaces but also to surfaces with arbitrary topology. In addition theoretically n-sided polygons such as pentagons or hexagons ay be included in the polygonal surface. Let be the position of the i-th node Let n be the nuber of nodes Let Vi be the i-th scattered point Let l be the nuber of scattered points While( terination condition is not satisfied ){ For( all nodes (i =... n) ){ Step : Calculate pseudo-noral For( all scattered points Vi (i =... l) ){ Step 2: Make connection of each scattered point to its nearest node For( all nodes (i =... n) ){ Step 3: Calculate displaceent d caused by the force exerted by V-Spring Step 4: Calculate displaceent u i caused by the force for regularizing node distribution Step 5: Calculate displaceent c i caused by the force exerted by scattered points Step 6: = + (d + u i + c i) Figure 5: Overview of the fitting algorith. (A) Q Q2 Figure 6: Least-square fitting of a planar curve to scattered points. (A) Geoetric interpretation of general least-square fitting. (B) Geoetric interpretation of our fitting approach. 4.6 Terination Condition To deterine when to terinate iterations the axiu aong the nors of all node displaceents is copared with a given threshold e. If the axiu nor is less than the threshold e the iterations are terinated. The size of the displaceent depends on the resolution of the V (B) V2 Q V

6 polygonal surface; therefore the displaceent should be noralized by the size of polygonal faces. In our ipleentation we noralize the displaceent of each node by the average length of its neighboring edges. Then the axiu nor of the noralized displaceents is copared with the threshold e. 4.7 Perforance Iproveent In iterative approaches of the Jacobi or Gauss-Seidel types high frequencies tend to be reoved quickly while it takes any iterations to reove low frequencies. Consequently if the nuber of nodes is extreely large it takes any iterations to achieve convergence. A proising way to reduce the execution tie is to eploy ulti-grid ethods [2][9][8]. Kobbelt and his colleagues [3] are positively using the ulti-grid ethod to odel dense eshes. By using the ulti-grid ethod a linear execution tie can be achieved. Multi-grid ethod requires polygonal surfaces with several different levels of resolutions. Mesh siplification algoriths [][8][2] [2][24] can provide the polygonal surfaces with ulti-resolutions. For exaple let M M2 and M3 be three levels of polygonal surfaces where M is the finest and M3 is the coarsest. The V-cycle ulti-grid ethod applies iterations for the polygonal surfaces with different levels in the sequence {M M2 M3 M2 M. The first-half process going down fro M to M2 is called pre-soothing and the second-half process coing up fro M3 to M is called post-soothing. In the pre-soothing soe iterations are perfored at each level in order to reove noise. On the coarsest level M3 a rough shape is predicted by the solution. As the post-soothing proceeds the rough shape approaches the precise shape. In the post-soothing iterations are perfored at each level until the terination condition described in Section 4.6 is satisfied. Our terination condition is noralized by the resolution; this provides an efficient way of deterining the tie at which to ove on. 4.8 Extension to Curve Modeling Basically the algorith for curve fairing is the sae as for surface fairing. In each iteration the node is oved to a new position by forces exerted by two neighboring nodes - and + (see Figure 4(A)). In the case of a planar curve there is no extended atter fro in the surface case; however in the case of a non-planar curve the calculation of the pseudo-noral is ore difficult than in the surface case. Fro a sequence of nodes - and + we calculate the unit tangent Ti and the unit binoral Bi as follows: Ti = Bi = ( + ) + ( ) ( + ) ( ) ( + ) where denotes the outer product. As the outer product of Bi and Ti we calculate the unit principal noral as follows: N i = Bi Ti Bi Ti. If - and + are collinear Bi and are zero vectors; then the displaceent d is a zero vector according to Equation (5). The best way to obtain a fair curve is to apply V-Spring forces in both directions of Bi and ; however in practice applying a force only in the direction of gives a fair curve even if the proble is a non-planar case. 5. Results Figure 7 shows a result of fair surface generation by using our algorith. Figure 7(A) shows the initial esh with sharp corners that is sudden change in curvature while the figures (B) and (C) show the faired results with two different sets of direct constraints. As seen in the figures the algorith produced fairer surfaces. The surface of Figure 7(B) resulted when positions of nodes on the inner- and outer-boundaries are kept unchanged by using the positional constraints. The surface of Figure 7(C) is generated by constraining norals of the nodes on the inner- and outer-boundaries in addition to positions of these nodes. Figure 9 shows another result of fairing in shaded iage. Figure 8 shows the stability of our algorith by giving it an extreely noisy esh as initial data. We added rando noise of large aplitude to the esh of Figure 7(A) then processed the noisy esh by using our algorith. Five iterations of pre-soothing generated a soother surface of Figure 8(B). Then the esh converged to a fair surface of Figure 8(C) with further processing. Table shows the execution tie of our algorith easured for eshes of various resolutions. It says that the tie grows approxiately linearly to the coplexity of given eshes. Table : Execution tie for fair surface generation. The execution tie is easured for the surface shape with the constraints in Figure 7(C). Colun (a) in the table is data for the resolution of Figure 7(C). Colun (b) (c) and (d) are data for different resolutions with the sae surface shape. These data are easured under the following conditions: CPU: PentiuII 450MHz Syste: WindowsNT4.0 Threshold e for terination condition: 0.00 Nuber of level for ulti-grid: 6 Nuber of iterations at each level of pre-soothing: 5. (a) (b) (c) (d) Nuber of nodes Nuber of faces (triangles) Execution tie (sec) Figure 0 shows results of least-square fitting of polygonal curves by using indirect constraints as described in Section 4.5. An initial curve shown by a noisy thin line converged to a sooth curve shown by a thick line. Node positions of the initial curve are assigned to scattered points. In Figure 0(A) all weights kj in Equation () used to take the trade-off between fairing and keeping proxiity are set to 0.0 while in Figure 0(B) they are set to As seen in the figures the algorith produced fairer curves under the indirect constraints. 6. Suary This paper presented an algorith to generate fair polygonal curves and surfaces based on iterative approach by using a new discrete spring odel. The algorith produced polygonal curves and surfaces whose local variation in curvature is iniized. In our discrete spring odel a linear spring which length approxiately represents a curvature is attached along the noral line of each node. Energy is assigned to the difference of the lengths that is difference in curvature of nearby springs. Our algorith then tries to iniize total energy by iterative approach. The algorith accepts various constraints 6

7 such as positional noral and least-square constraints so that useful surface odels can be generated for coputer graphics coputer aided design and other geoetric odeling applications. The ipleentation of the algorith is easy due to its geoetrically intuitive interpretation. The results of experients showed that the algorith generated fair surfaces that satisfy positional noral and other constraints. The algorith was robust under the presence of positional noise in the initial polygonal data. It also exhibited that coputational costs increased approxiately linear to the nuber of nodes consisting in a polygonal curve (surface). In our experience the algorith is significantly robust and stable; however the convergence of the algorith should be proved in the future work. The target of this paper is to apply our spring odel to shape odeling; however fairing proble is not liited to the application. Applying our spring odel to the other applications is also the future work. References [] N. Aenta M. Bern and M. Kavysselis A New Voronoi-Based Surface Reconstruction Algorith Coputer Graphics (SIGGRAPH'98 Conference Proceedings) pp [2] W. Briggs A Multi-grid Tutorial SIAM Philadelphia 977. [3] M. P. Do Caro Differential Geoetry of Curves and Surfaces Prentice Hall 976. [4] E. Catull and J. Clark Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes Coputer Aided Design 0(6) pp [5] G. Celniker and D. Gossard Deforable Curve and Surface Finite-Eleents for Free-For Shape Design Coputer Graphics (SIGGRAPH'9 Conference Proceedings) pp [6] D. DeCarlo D. Metaxas and M. Stone An Anthropoetric Face Model using Variational Techniques Coputer Graphics (SIGGRAPH '98 Conference Proceedings) pp [7] M. Eck and H. Hoppe Autoatic Reconstruction of B-Spline Surfaces of Arbitrary Topological Type Coputer Graphics (SIGGRAPH '96 Conference Proceedings) pp [8] M. Garland and P. S. Heckbert Surface Siplification Using Quadric Error Metrics Coputer Graphics (SIGGRAPH '97 Conference Proceedings) pp [9] W. Hackbusch Multi-Grid Methods and Applications Springer-Verlag Berlin New York 985. [0] M. Halstead M. Kass and T. DeRose Efficient Fair Interpolation using Catull-Clark Surfaces Coputer Graphics (SIGGRAPH '93 Conference Proceedings) pp [] K. Ho-Le Finite Eleent Mesh Generation Methods: A Review and Classification Coputer Aided Design 20() pp [2] H. Hoppe Progressive Meshes Coputer Graphics (SIGGRAPH '96 Conference Proceedings) pp [3] L. Kobbelt S. Capagna J. Vorsatz and H. P. Seidel Interactive Multi-Resolution Modeling on Arbitrary Meshes Coputer Graphics (SIGGRAPH '98 Conference Proceedings) pp [4] R. M. Koch M. H. Gross F. R. Carls D. F. von Buren G. Fankhauser and Y. I. H. Parish Siulating Facial Surgery Using Finite Eleent Methods Coputer Graphics (SIGGRAPH '96 Conference Proceedings) pp [5] B. Levy and J. L. Mallet Non-Distorted Texture Mapping for Sheared Triangulated Meshes Coputer Graphics (SIGGRAPH '98 Conference Proceedings) pp [6] J. L. Mallet Discrete Sooth Interpolation ACM Transactions on Graphics 8(2) pp [7] J. L. Mallet Discrete Sooth Interpolation in Geoetric Modelling Coputer Aided Design 24(4) 992 pp [8] C. McCorick Multilevel Adaptive Methods for Partial Differential Equations SIAM Philadelphia 989. [9] H. P. Moreton and C. H. Sequin Functional Optiization for Fair Surface Design Coputer Graphics (SIGGRAPH '92 Conference Proceedings) pp [20] D. F. Rogers and N. G. Fog Constrained B-Spline Curve and Surface Fitting Coputer Aided Design 2(0) pp [2] W. J. Schroeder J. A. Zarge and W. E. Lorensen Deciation of Triangle Meshes Coputer Graphics (SIGGRAPH '92 Conference Proceedings) pp [22] R. Szeliski and D. Tonnesen Surface Modeling with Oriented Particle Systes Coputer Graphics (SIGGRAPH '92 Conference Proceedings) pp [23] G. Taubin A Signal Processing Approach to Fair Surface Design Coputer Graphics (SIGGRAPH '95 Conference Proceedings) pp [24] G. Turk Re-Tiling Polygonal Surfaces Coputer Graphics (SIGGRAPH '92 Conference Proceedings) pp [25] W. Welch and A. Witkin Variational Surface Modeling Coputer Graphics (SIGGRAPH '92 Conference Proceedings) pp [26] W. Welch and A. Witkin Free-For Shape Design Using Triangulated Surfaces Coputer Graphics (SIGGRAPH '94 Conference Proceedings) pp [27] O. C. Zienkiewicz The Finite Eleent Method Third Edition McGraw-Hill United Kingdo 977. Figure 7: (A) Initial esh. (B) Defored result fro (A) under the positional constraints on inner- and outer-boundary nodes. (C) Defored result fro (A) under the positional and noral constraints on inner- and outer-boundary nodes. Arrows on boundary nodes show the directions of noral constraints. 7

8 Figure 8: Stability of the algorith. (A) Initial esh with rando noise on the nodes. (B) Shape (A) after five iterations of pre-soothing on the finest level. (C) Shape (A) after full convergence. Figure 9: Shaded iage of a defored result. (A) Initial esh. (B) Defored result fro (A) under the positional and noral constraints on nodes of upper- lower- and inner-boundaries. Figure 0: Least-square fitting of a polygonal segent to scattered points. A noisy thin line represents an initial segent. A sooth thick line represents a converged segent. Node positions of the initial segent are given as scattered points. (A) All weights kj in Equation () are set to 0.0. (B) All weights kj are set to