G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. SIGGRAPH 91 (1991) 257266.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....or monotonic. Applying constraints when fairing data is a similar application. A recent collection of papers summarises much work in both areas [10] Local modifications of single surfaces may use constraints on control (or other) points during user modification of the shape of the surface [5, 25]. Such approaches are often referred to as physics based modelling : changes in shape of a dynamically deformable surface are controlled by a set of virtual springs attached to it, constraining its shape [20] A survey is provided by Gibson [8] Later work has included other constraints such as ....

G. Celniker and D. Gossard, "Deformable curve and surface finite-elements for free-form shape design", Computer Graphics (SIGGRAPH 91), Vol 25, No 4, 1991, pp. 257--266 29

....geometric deformation approaches, there are physics based deformable models. A physicsbased model employs elasticity theory to construct an energy functional, and calculates the model s shape directly by finding a minimum to the energy functional or by solving a set of differential equations [6, 16]. Physics based models can most closely mimic the sculpting effect and their optimization algorithms automatically produce fair surfaces. However, they typically require intensive computations. To produce curved surfaces from polygonal shapes, a traditional method is to round sharp corners and ....

G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. Computer Graphics, 25(4):257--266, 1991.

.... carried out either by directly moving groups of surface control points or by deforming the space in which surface control points are embedded [37] 38] 9] Other surface deformation schemes assign physical prop erties to surfaces so that the surfaces may be deformed by external forces [34] 42][5]. The behavior of the aforementioned surface deformation schemes all depends on the density and connectivity of surface control points. The user, as a result, must actively maintain the density and connectivity of the surface control points in order to obtain the desir able surface deformations. ....

Celniker, G., and Gossard, D, "Deformable curve and surface finite-elements for free-form shape design", Proceedings of SIGGtAPH '9I, pp. 257-266 (1991).

....flat. Consequently, first and second order derivatives decouple and the associated energy minimization functional is reduced to a simple linear combination of weighted first and second derivative squared norms of the surface. While this formulation is very useful for variational geometric modeling [8, 11, 31] and intuitive direct manipulation of surfaces [24, 23, 27] it cannot capture subtleties of the nonlinear dynamic behavior of more complex shapes. These nonlinearities are particularly important for the accurate modeling of stability phenomena, which occur, among other situations in the ....

CELNIKER, G., AND GOSSARD, D. Deformable Curve and Surface Finite Elements for Free-Form Shape Design. Computer Graphics (Proceedings of SIGGRAPH 91) 25, 4 (1991), 257--266.

....an elaborate validation and error analysis with respect to craniofacial surgery. 1. 3 Our Approach To optimize both accuracy and rendering quality our goal was to combine the physical correctness of volumetric finite element simulation with the superior quality of the C continuous surface of [4, 10]. Furthermore, a validation of the proposed model will investigate its applicability to facial surgery simulation. As a first major contribution we therefore extended the surface based approach of [10] to volumetric physics which involved both the reformulation of the mathematical and physical ....

....are depicted in figure 7. In the following sections we step by step derive a set of twelve functions that feature the required C prism surface. The major advantage of these functions compared to a straightforward tensor product extension of the triangle surface shape functions presented in [4] and [10] is that they achieve the desired smoothness at the surface with only nine DOFs instead of twelve. Trivariate C Shape Functions Let R, S, and T define a barycentric surface coordinate system with and let Q denote the volumetric extension with at the top surface and at the bottom of ....

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G.Celniker and D.Gossard. "Deformable curve and surface finite elements for free-form shape design." In T.W. Sederberg, editor, Computer Graphics (SIGGRAPH '91 Proceedings), volume25, pages 257--266, July 1991.

.... for deformable parts [56, 57] A survey of deformable modeling in computer graphics can be found in [22] The use of physical simulation and related optimization techniques as a means of geometric interaction has been applied to animation [58] drawing [59] free form surface and volume modeling [15], mechanical design [65] and interactive molecular simulation [55] For a discussion on the dynamic simulation of nonpenetrating flexible bodies see [6] Models and algorithms appropriate for the collision of deformable bodies are investigated in [19] 4 An Algorithm for Planning Paths for ....

G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. Computer Graphics ($IGGRAPH'91), pages 257-266, 1991.

....is currently under investigation. In graphics physically based models have been proposed for deformable parts [19, 20] The use of physical simulation and related optimization techniques as a means of geometric interaction has been applied to animation [21] free form surface and volume modeling [4], and mechanical design [23] For a discussion on the dynamic simulation of non penetrating flexible bodies see [1] 3 f PRM: General Description f PRM repeats a basic step until a query is answered or until an predefined amount of time has elapsed. The method can be seen as a single shot method ....

G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. Com- puter Graphics (SIGGRAPH'91), pages 257-266, 1991.

....et al. 32] to achieve interactive simulations of a virtual liver composed of about 2000 tetrahedra. But we are interested in interactively deforming objects that cannot be approximated well by so few tetrahedra. Deformable surfaces have also been used for geometric modeling. Celniker and Gossard [9] applied the finite element method to minimize surface energy while meeting constraints. These ideas were later extended to NURBS by Terzopoulos and Qin [43] and to Catmull Clark subdivision surfaces by Qin et al. 36] For some applications dynamic motion has not been deemed necessary so static ....

George Celniker and Dave Gossard. Deformable curve and surface finiteelements for free-form shape design. Computer Graphics, 25(4):257--266, July 1991.

....energy minimization, with control net (surface patches lying substantially in the St. Lawrence River are darkly coloured) The vertical scale has been exaggerated 40 times to enhance detail. faces, an appropriate smoothing functional is the linearized thin plate energy functional J(F (x; y) [Gre94a, Gre94b, CG91, HKD93], defined as J(F ) F xx 2F xy F yy dx dy: 6) The region Omega can be used to restrict the functional to only part of the surface in question, thereby localizing the smoothing effect. Figure 1 shows a truncated cone which has been capped in such a way as to minimize the thin ....

G. Celniker and D. Gossard. Deformable curve and surface finite-elements for freeform shape design. In Proceedings of SIGGRAPH '91, pages 257--265. ACM

....mesh is just a special case of FEM. Its accuracy and mathematical rigorousness make FEM a better choice for applications such as surgical simulations. Terzopoulos et al.[21, 20, 22] applies both finite difference and finite element methods in modeling elastically deformable objects. Celniker et al.[15] applies FEM to generate primitives that build continuous deformable shapes designed to support a new free form modeling paradigm. Pieper et al.[16] applies FEM to computer aided plastic surgery. Chen [3] animates human muscle using a 20 node hexahedral FEM mesh. Keeve et al.[11] develops a static ....

G. Celniker nad G. Gossard. Deformable curve and surface finite elements for free form shage design. Computer Graphics, 25(4), 1991.

....mesh is just a special case of FEM. Its accuracy and mathematical rigorousness make FEM a better choice for applications such as surgical simulations. Terzopoulos et al.[29, 28, 30] applies both finite difference and finite element methods in modeling elastically deformable objects. Celniker et al.[15] applies FEM to generate primitives that build continuous deformable shapes designed to support a new free form modeling paradigm. Pieper et al.[17] applies FEM to computer aided plastic surgery. Chen [5] animates human muscle using a 20 node hexahedral FEM mesh. Keeve et al.[10] develops a static ....

G. Celniker nad G. Gossard. Deformable curve and surface finite elements for free form shage design. Computer Graphics, 25(4), 1991.

....mesh is just a special case of FEM. Its accuracy and mathematical rigorousness makes FEM a better choice for applications such as surgical simulations. Terzopoulos et al.[12, 11, 13] applies both finite difference and finite element methods in modeling elastically deformable objects. Celniker et al.[6] applies FEM to generate primitives that build continuous deformable shapes designed to support a new free form modeling paradigm. Pieper et al.[8] applies FEM to computer aided plastic surgery. Chen [3] animates human muscle using a 20 node hexahedral FEM mesh. Keeve et al.[5] develops a static ....

G. Celniker nad G. Gossard. Deformable curve and surface finite elements for free form shage design. Computer Graphics, 25(4), 1991.

....of data points. Many techniques have been developed for this purpose [18, 13] the most widely used ones being based on piecewise polynomials. However, to obtain high quality curves differentiability is usually not sufficient but optimality with respect to some fairness measure is required as well [3, 26]. Applications such as level of detail rendering [25] data compression, progressive transmission, hierarchical editing [14] and adaptive numerical solvers [16, 24] require widely varying levels of detail. This makes it desirable to have multiresolution representations and associated wavelets for ....

....Note that the subdivision scheme itself is not affected by this modification since for pure subdivision the odd wire carries only zeroes (see Figure 13) 3 Although in some cases the instability is so weak that for all practical purposes no ill effects appear. Scheme columns of V K 2 1 64 [ 3,15,15, 3] T K 3 1 512 [ 25,105,105, 25] T K spl 1 144 [ 7,31,31, 7] T K 4pt 1 32 [ 1,9,9, 1] T Table 1: Lifting weights for variational subdivision schemes leading to stable transforms in the uniform setting. These weights were computed by considering the first 4 moments of the scaling functions ....

CELNIKER,G.,AND GOSSARD, D. Deformable Curve and Surface Finite Elements for Free-Form Shape Design. Computer Graphics 25 (1991), 257--265.

....evaluation and minimization of these expressions would be computationally too expensive. Therefore they are approximated with the following ones: E bend (x) Z kx 00 (t)k 2 dt; 5) E stretch (x) Z kx 0 (t)k 2 dt: 6) 5 These approximations are frequently used (see for instance [Celniker and Gossard, 91] The advantage of these approximations is that for e.g. B spline curves they are quadratic functions of the control points (see section 5) This is probably the reason for their popularity, since quadratic functions can be efficiently minimized. A disadvantage of the approximations (5) and (6) ....

Celniker, G. and Gossard, D. (1991), Deformable curve and surface finite-elements for free-form shape design, Computer Graphics 25, 257--266.

....Section 8 concludes the article. 2 Background Dynamic NURBS are motivated by prior research aimed at applying the deformable modeling approach to shape design. Terzopoulos and Fleischer [36] demonstrated simple interactive sculpting using viscoelastic and plastic models. Celniker and Gossard [5] developed an interesting prototype system for interactive free form design based on the finite element optimization of energy functionals proposed in [36] Bloor and Wilson [3] developed related models using similar energies and numerical optimization, and in [2] they proposed the use of ....

....function. The dissipation energy is F = 1 2 Z Z fl s s du dv = 1 2 p D p; 13) where D(p) Z Z flJ J du dv (14) is the damping matrix. For the elastic potential energy of D NURBS, we can adopt the thin plate under tension energy model [35] which was also used in [5, 42] (other energies are possible, including the nonquadratic, curvature based Published in ACM Transactions on Graphics, 13(2) April, 1994, 103 136. 8 energies in [36, 19] U = 1 2 Z Z ff 1;1 s u s u ff 2;2 s v s v fi 1;1 2 s u 2 2 s u 2 fi ....

G. Celniker and D. Gossard. Deformable curve and surface finite-element for free-form shape design. Computer Graphics, 25(4):257--266, 1991.

....cannot be achieved without realistic simulation techniques for such objects. Since Terzopoulos[10] characterized the simulation of cloth as a problem in deformable surface, many research groups have tried to give efficient solution to the cloth simulation, and various techniques have been proposed[1, 4, 2, 3, 11, 12]. Most of the techniques are based on physically based modeling, and the problem is formulated as an ordinary differential equation[1] Explicit Euler integration is the simplest approach to this problem. This approach calculates the internal forces at the current state and derives the next state ....

G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. Computer Graphics(Proc. of SIGGRAPH '91), pages 257--266, 1991.

....Background D NURBS are motivated by prior research aimed at applying the deformable modeling approach to shape design. Terzopoulos and Fleischer demonstrated simple interactive sculpting using viscoelastic and plastic models [33] Celniker and Gossard developed an interesting prototype system [5] for interactive free form design based on the finite element optimization of energy functionals proposed in [33] The system combines geometric constraints with sculpting operations based on forces and loads to yield fair shapes. However, this approach does not provide interactive mechanisms in ....

....= Z Z flJ J du dv (16) 7 where (u; v) is the prescribed mass density function over the parametric domain of the surface and fl(u; v) is the prescribed damping density function. To define an elastic potential energy for the surface, we adopt the thin plate under tension energy model [32] [5], 38] 16] 35] This yields the N Theta N stiffness matrix K(p) Z Z Gamma ff 1;1 J u J u ff 2;2 J v J v fi 1;1 J uu J uu (17) fi 1;2 J uv J uv fi 2;2 J vv J vv Delta du dv; where the subscripts on J denote parametric partial derivatives. The ff i;j (u; ....

G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. Computer Graphics, 25(4):257--266, 1991. (Proc. ACM Siggraph'91).

....dynamic behavior governed by the physical laws of elasticity and demonstrated simple interactive sculpting using viscoelastic and plastic models. Bloor and Wilson [3] demonstrated free form design using tensor product B splines and the optimization of energy functionals. Celniker and Gossard [4] developed an interesting prototype system for interactive design based on surface finite elements. Welch and Witkin [30] made similar use of trimmed hierarchical B splines. Moreton and Sequin [18] interpolated a minimum energy curve network with quintic Bezier patches by minimizing the variation ....

.... [10] We express the kinetic energy due to a prescribed mass distribution function (u; v) over the parametric domain of the surface and a dissipation energy due to a damping density function fl(u; v) To define an elastic potential energy, we adopt the thin plate under tension energy model [26, 4, 30] U = 1 2 Z Z i ff 1;1 s 2 u ff 2;2 s 2 v fi 1;1 s 2 uu fi 1;2 s 2 uv fi 2;2 s 2 vv j du dv: The subscripts on s denote parametric partial derivatives. The ff i;j (u; v) and fi i;j (u; v) are elasticity functions which control tension and rigidity, respectively. Other ....

G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. Computer Graphics, 25(4):257--266, 1991. (Proc. ACM Siggraph'91).

....second, post processing step, one has to smooth these regions. This procedure is called fairing (see [12] for basics on the subject) In CAGD one approach has gained a lot of interest recently. It is called called variational design and can handle these kind of problems very efficiently (see e.g. [1, 2, 3, 7, 8, 11, 13, 15, 16, 18]) In the present paper we want to describe the basic ideas and will explain how this method can be applied to the above mentioned problems. The paper is organized as follows. In Sec. 2 we describe the general procedure, and explain which kind of constraints are permissible. In Sec. 3 the problem ....

....functionals J 2 and J 4 have been investigated by Moreton Sequin. They used them to design closed surfaces (of positive genus) having optimal shape (see [15, 16] The following more general quadratic second order functional is used by Welch Witkin (see [18] and Celniker Gossard (see [3]) J : G 7 Z Omega P 2 i;j=1 ff ij D G u i fi fi fi G u j E fi ij i 2 G u i u j j 2 du 1 du 2 : 3 meaning order exact simplification area functional 1 J 1 : G 7 R Omega 1 d G J 6 : G 7 R Omega i G u1 j 2 i G u2 j 2 du 1 du 2 thin plate ....

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G. Celniker and D. Gossard. Deformable curve and surface finite--element for free--form shape design. ACM Computer Graphics, 25:257--266, 1991.

....provide a means of overcoming these difficulties. It is possible to construct free form surfaces with natural dynamic behavior governed by physical laws [8, 9, 10] Terzopoulos and Fleischer [11] demonstrated simple interactive sculpting using viscoelastic and plastic models. Celniker and Gossard [12] developed an interesting prototype system for interactive free form design based on this idea and the finite element optimization of energy functionals. Bloor and Wilson [13] used similar energies optimized through numerical methods and they employed B splines for this purpose. Subsequently, ....

.... prescribed mass distribution function (u; v) over the parametric domain of the surface and a Raleigh dissipation energy due to a damping density function fl(u; v) To define an elastic potential energy, we adopt the thin plate under tension energy model which was proposed in [18] and also used in [12, 14, 17, 19] U = 1 2 Z Z ff 1;1 s u s u ff 2;2 s v s v fi 1;1 2 s u 2 2 s u 2 fi 1;2 2 s u v 2 s u v fi 2;2 2 s v 2 2 s v 2 du dv: 11) The ff i;j (u; v) and fi i;j (u; v) are elasticity functions which control ....

G. Celniker and D. Gossard. Deformable curve and surface finite-element for free-form shape design. Computer Graphics, 25(4):257--266, 1991.

....that requires continuity of parametric derivatives (so called parametric continuity) is the inability to model surfaces of arbitrary topological type (cf. Herron [8] It is not possible, for instance, to model a sphere or a deformed sphere using a Clough Tocher interpolant. Celniker and Gossard [3] recently presented an interpolation method that extends Clough Tocher interpolation by setting the remaining degrees of freedom so as to minimize a fairness norm. The fairness norm they use is quadratic, so it can be minimized by solving a (sparse) linear system. As a result, their method is fast ....

....we introduce additional degrees of freedom into the surface by subdivision, and then set the degrees of freedom by optimizing a fairness norm on the surface subject to a set of linear constraints given by the interpolation conditions. 4. 1 Evaluating the Fairness Norm Celniker and Gossard [3] were able to improve the quality of interpolating surfaces using a fairness norm based on a linear combination of the energy of a membrane and a thin plate. Without any fundamental changes, the norm can be given directional preferences and nonuniform weighting over the surface, but for clarity of ....

George Celniker and Dave Gossard. Deformable curve and surface finite elements for free-form shape design. In Proceedings of SIGGRAPH '91, pages 257--265, July 1991.

....and addresses the issue of system assessment. Section 7 concludes the paper and outlines future research directions. 2 PHYSICS BASED MODELING Various techniques have been developed to generate fair surfaces that satisfy multiple constraints and optimize an energy based objective functional [3, 15, 33]. It is also possible to construct dynamic surfaces with natural behavior governed by physical laws [17, 30] The benefit of physics based behavior during interactive design is that the development of the surface follows intuitive physical paths and the surfaces react to external manipulation in a ....

....aids the design task, especially with the reality based feedback obtained from the haptic device. Terzopoulos and Fleischer [29] provided a basis for physics based design by combining the two techniques for simple interactive sculpting using viscoelastic and plastic models. Celniker and Gossard [3] developed a prototype system for interactive design based on the finite element optimization of energy functionals. Bloor and Wilson [2] used similar energies optimized through numerical methods for B splines. Moreton and S# equin [15] interpolated a minimum energy curve network with quintic ....

G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. In Computer Graphics (SIGGRAPH '91 Proceedings), volume 25, pages 257--266, July 1991.

....be approximated to any user specified error tolerance, making it useful for simultaneous graphics rendering as well as haptic rendering. 3 RELATED WORK Various methods have been developed to generate fair surfaces which satisfy multiple constraints and optimize an energy based objective function [2, 11, 27]. It is also possible to construct dynamic surfaces with natural behavior governed by physical laws [12, 24] Terzopoulos and Fleischer [23] demonstrated simple interactive sculpting using viscoelastic and plastic models. Celniker and Gossard [2] developed an interesting prototype system for ....

....optimize an energy based objective function [2, 11, 27] It is also possible to construct dynamic surfaces with natural behavior governed by physical laws [12, 24] Terzopoulos and Fleischer [23] demonstrated simple interactive sculpting using viscoelastic and plastic models. Celniker and Gossard [2] developed an interesting prototype system for interactive design based on the finite element optimization of energy functionals. Bloor and Wilson [1] used similar energies optimized through numerical methods for B splines. Celniker and Welch [3] investigated deformable B splines with linear ....

G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shapedesign. In Computer Graphics (SIGGRAPH '91 Proceedings), volume 25, pages 257-- 266, July 1991.

....design, polygonal models, energy minimization 1. Introduction The purpose of this paper is to provide an algorithm for generating fair curves and surfaces for use in the fields of computer graphics (CG) and computer aided design (CAD) Generation of fair shapes is a major topic in shape design [5][11] 19] 21] 25] 26] 27] 29] It is also required in applications such as fitting of smooth shapes to scattered points [8] 23] texture mapping [16] and so on. The curves and surfaces treated in this paper are represented in polygonal form. The inputs of the algorithm are (1) an initial ....

....approaches, finite element approaches generally yield a higher quality surface and generate an explicit surface equation for each face, while they are ###########87962 692432##062607 ######87962 ## ##9442 # 1 computationally more expensive. Celniker and Gossard s approach [5] generates a C1continuous surface by using a finite element technique. They applied Zienkiewicz s shape function [30] which was originally proposed for finite element analysis, to attain C1 continuity in surface modeling. Their approach minimizes a weighted combination of the energy factors of a ....

G. Celniker and D. Gossard, Deformable Curve and Surface Finite-Elements for Free-Form Shape Design, Computer Graphics (SIGGRAPH'91 Conference Proceedings), pp. 257-266, 1991.

....the vertices of the polyhedral surface without changing the connectivity of the faces. The faired surface has exactly the same number of vertices and faces as the original one. However, our signal processing formulation results in much less expensive computations. In these variational formulations [5, 24, 38, 12], after finite element discretization, the problem is often reduced to the solution of a large sparse linear system, or a more expensive global optimization problem. Large sparse linear systems are solved using iterative methods [9] and usually result in quadratic time complexity algorithms. In ....

G. Celniker and D. Gossard. Deformable curve and surface finiteelements for free-form shape design. Computer Graphics, pages 257-- 266, July 1991. (Proceedings SIGGRAPH'91).

....is minimized. The idea is to compute a mesh which is as smooth as possible while still containing a controllable amount of geometric detail. Fig. 10 shows an example. From CAGD it is well known that constrained energy minimization is a very powerful technique to generate high quality surfaces [3, 14, 28, 30, 37]. For efficiency, one usually defines a simple quadratic energy functional f and searches among the set of functions satisfying prescribed interpolation constraints for that function f which minimizes . Transferring the continuous concept of energy minimization to the discrete ....

CELNIKER, G., AND GOSSARD, D. Deformable curve and surface finite elements for free-form shape design. In Computer Graphics (SIGGRAPH 91 Proceedings) (July 1991), pp. 257--265.

....on Navier s equation) can be tuned using a few, well known material properties; incompressibility is trivial to specify using Poisson s ratio # = 1 2 . The Finite Element Method (FEM) a widely used, flexible and accurate method for solving Navier s equations, has seen application in graphics [13, 7, 8, 6]. However, its use has been limited by the complexity of the method and the cost of solving the resulting linear system. While FEM techniques are e#ective general tools for (especially nonlinear) elasticity, we believe that the BEM, on which our technique is based, has certain features which will ....

George Celniker and Dave Gossard. Deformable curve and surface finite elements for free-form shape design. In Thomas W. Sederberg, editor, Computer Graphics (SIGGRAPH 91 Conference Proceedings), volume 25, pages 257--266, July 1991.

....dense, square matrix. Then they solve a set of linear equations to compute the positions of the control points of a second mesh whose limit Catmull Clark surface interpolates the original mesh. In addition, they derived and applied a fairness norm based on Celniker and Gossard s fairness norm [CG91] Since interpolating algorithms often produce undesirable undulations in limit curves and surfaces, a fairing or smoothing process is sometimes used. Such processes typically achieve this result by minimizing some energy functional that in turn minimizes arc lengths or curvature norms. Levin ....

....together at their boundaries to create more complex objects. Their algorithm uses an energy minimization function whose purpose is to preserve the volume during sculpting. The derivation of the functional, although lengthy, is fairly straightforward and is similar to the energy functional used in [CG91, TQ94] In addition to the volume preserving constraint, their system can satisfy interpatch continuity constraints, positional constraints, attachment constraints, and inter point constraints. All of these are formulated using a Lagrange multiplier method (see Section 7.3) 8 s(u, v, w) # ....

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G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. In Computer Graphics (SIGGRAPH 91 Proceedings), pages 257--266, July 1991.

....to the modeling and computer graphics communities by Terzopoulos and colleagues [24, 23, 22, 14] In a nutshell, the geometry of their models is discretized from continuous surfaces and solids and is attached to mass spring meshes. Others have used modal dynamics [17] and the finite element method [5] in order to improve the stability and accuracy of dynamic models. Baraff, Witkin, Kass and others [27, 2, 3] have developed techniques for animating and constraining non penetrating dynamic surfaces. Qin and colleagues [20, 19, 12] derived dynamic models for direct manipulation of spline and ....

G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. In Computer Graphics (SIGGRAPH 91 Proceedings), pages 257--266, July 1991.

....uu x uv 2fi 11 fi 22 x uu x vv 4fi 2 12 x uv 2 4fi 12 fi 22 x uv x vv fi 2 22 x vv 2 1 C C C C A du dv: 2:8) This functional is highly nonlinear in the vector and matrix norms, and leads to a difficult nonlinear optimization problem. It is therefore common [welc92] cari92] [celn91] [terz87] to simplify the functional by linearizing the matrix norms and B to produce the thin plate under tension model [schw66] ZZ u;v i kGk ff fi 11 x uu 2 2fi 12 x uv 2 fi 22 x vv 2 j dudv: 2:9) This approximation is only accurate near the actual minimum physically it is ....

....deformed surface, they discretized Equation 2.12 using finite differences to form equations approximating the stress and strain energies. By applying external forces to the resulting equations and numerically integrating through time, the behavior of a surface is simulated. Celniker and Gossard [celn91] applied a weighted sum of continuous shape functions from finite element theory to approximate the thin plate under tension functional (Equation 2.9) This approximation was used to set the remaining degrees of freedom of triangular surface patches constrained by both geometric constraints (input ....

George Celniker and Dave Gossard. "Deformable curve and surface finite-elements for free-form shape design". Computer Graphics (SIGGRAPH '91 Proceedings), Vol. 25, No. 4, pp. 257--266, July 1991.

....makes representing and controlling such shapes on a computer a difficult problem. 1. 1 Functional minimization for shape design Optimization has long been used as a way of describing fair freeform shapes (a good survey is Moreton[23] More recently, it has come to be used in interactive modelers[4,5,41,18]. Though such approaches are computationally complex, their intent is to create an illusion of simplicity for the designer. Ideally, the designer sees a surface having no particular fixed controls or other representationspecific parameters. Instead, the surface can be directly manipulated, pinned ....

....down at points and along curves, and will behave as if made of some infinitely stretchy material. This lets us mimic a style of pen and paper design in which important contours of a shape are sketched out as character lines , with the understanding that a surface passes through them in a fair way[4]. Such shapes are ultimately realized as solutions to constrained functional minimization problems globally fair surfaces that satisfy geometric interpolation constraints. This approach allows concise descriptions of a useful class of free form shapes. Unfortunately, these approaches have ....

[Article contains additional citation context not shown here]

George Celniker and Dave Gossard. Deformable curve and surface finite-elements for free-form shape design. Computer Graphics, 25(4), July 1991. (Proceedings Siggraph '91).

....to the underlying representation. In an attempt to move away from this paradigm, work on variational shape design attempts to provide a more abstract level of control over the shape to the designer, such as construct a smooth surface which passes through these points and contains this curve [CG91, GC95, HKD93, MS92, TQ94, WW92, WW94] This section provides a brief overview of variational modeling. Later in this document (in Chapter 3) the underlying method used for the automatic generation of varied face geometries will draw on the techniques presented here, using anthropometric ....

G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. In Proceedings SIGGRAPH '91, volume 25, pages 257--266, 1991.

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G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. SIGGRAPH 91 (1991) 257266.

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G. CELNIKER AND D. GOSSARD, Deformable curve and surface finite elements for free-form shape design, SIGGRAPH 91 (1991) 257-266.

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G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. In Proceedings of the 18th annual conference on Computer graphics and interactive techniques, pages 257--266. ACM Press, 1991.

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George Celniker and Dave Gossard. Deformable curve and surface finite elements for free-form shape design. In Thomas W. Sederberg, editor, Computer Graphics (SIGGRAPH 91 Conference Proceedings), volume 25, pages 257--266, July 1991.

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G Celniker and D Gossard. Deformable Curve and Surface Finite Elements for Free-form Shape Design. Computer Graphics, 25(4), 1991.

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George Celniker and Dave Gossard. Deformable curve and surface finite-elements for free-form shape design. In Computer Graphics (SIGGRAPH '91 Proceedings), volume 25, pages 257--266, July 1991.

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Celniker G., Gossard, D.: Deformable curve and surface finite-elements for free-form shape design. In ACM SIGGRAPH Conference Proceedings, 257-266 (1991).

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Celniker, G., and Gossard, D. Deformable curve and surface finiteelements for free-form shape design. Proceedings of SIGGRAPH'91 (Las Vegas, Nev., July 28--August 2, 1991.

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Celniker, G. and Gossard, D., "Deformable Curve and Surface Finite-Elements for FreeForm Shape Design", Computer Graphics, Vol. 25, No. 4, July 1991

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George Celniker and Dave Gossard. Deformable curve and surface finite elements for free-form shape design. In Thomas W. Sederberg, editor, Computer Graphics (SIGGRAPH 91 Conference Proceedings), volume 25, pages 257--266, July 1991.

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G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-from shape design. In Computer Graphics (SIGGRAPH '91 Proceedings), volume 25, pages 257--266, July 1991.

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George Celniker and Dave Gossard. Deformable curve and surface finite-elements for free-form shape design. Computer Graphics (SIGGRAPH 91), 25(4):257--266, July 1991.

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G. Celniker and D. Gossard. Deformable Curve and Surface Finite Elements for Free-Form Shape Design. In SIGGRAPH Comp. Graph. Proc., pages 257--265. ACM, 1991.

No context found.

G. Celniker and D. Gossard. Deformable curve and surface finite elements for free-form shape design. Computer Graphics (SIGGRAPH '91), 25:257-266, July 1991.

No context found.

G. Celniker and D. Gossard. Deformable curve and surface finite-element for free-form shape design. In ACM Computer Graphics (SIGGRAPH '91 Proceedings), pages 257--266, 1991.

No context found.

G. Celniker and D. Gossard. Deformable curve and surface finite-element for free-form shape design. In ACM Computer Graphics (SIGGRAPH '91 Proceedings), pages 257--266, 1991.

No context found.

G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. Computer Graphics, pages 257--266, July 1991. (Proceedings SIGGRAPH'91).

No context found.

CELNIKER,G.,AND GOSSARD, D. Deformable curve and surface finite-elements for free-from shape design. Computer Graphics 25, 4 (July 1991), 257--266.

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