## Free-Form Shape Design Using Triangulated Surfaces (1994)

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### BibTeX

@INPROCEEDINGS{Welch94free-formshape,

author = {William Welch and Andrew Witkin},

title = {Free-Form Shape Design Using Triangulated Surfaces},

booktitle = {},

year = {1994},

pages = {247--256}

}

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### Abstract

We present an approach to modeling with truly mutable yet completely controllable free-form surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a constrained shape optimization, with minimization of squared principal curvatures yielding graceful shapes that are free of the parameterization worries accompanying many patch-based approaches. Triangulated point sets are used to approximate these smooth variational surfaces, bridging the gap between patch-based and particle-based representations. Automatic refinement, mesh smoothing, and re-triangulation maintain a good computational mesh as the surface shape evolves, and give sample points and surface features much of the freedom to slide around in the surface that oriented particles enjoy. The resulting surface triangulations are constructed and maintained in real time. 1 Introduction ...

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(Show Context)
Citation Context ... these points. Prior to the fit, we cannot assign these coordinates based on arclength because we have no curve. So instead, we will make a chord-length approximation to an arc-length parameterization=-=[9,8]-=-, in which the parametric interval between two samples is taken to be the 3D Euclidean distance between them. In constructing a neighborhood parameterization for surface fitting, it is common to proje... |

812 |
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Citation Context ...se definition is independent of parameterization. Recall from the differential geometry of surfaces that information about the curvature of a surface at a point is given by the second fundamental form=-=[32]-=-. The normal section curvature of a surface s(u; v) in the direction of a parametric unit tangent t = [t u ; t v ] is given bys= II(t; t); where II(t; t) = t T s uu 1 n s uv 1 n s vu 1 n s vv 1 n t; (... |

411 | Re-tiling polygonal surfaces
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(Show Context)
Citation Context ...distribution We begin with the problem of maintaining a uniform sampling density over a surface. One approach that has been used successfully for both smooth and polyhedral surfaces is point repulsion=-=[29, 38,39,42]-=-: points move under the influence of mutual repulsive (or attractive) forces between points, constrained to act within the surface. In these schemes, a pair of points' influence on each other falls of... |

369 | Mesh Optimization
- Hoppe
(Show Context)
Citation Context ... apart. Similarly, if any node is too close to each of its neighbors, the node is destroyed using an edge-collapse operation. Both of these operations preserve the surface topology, and Hoppe, et. al.=-=[16]-=- showed that they are sufficient to transform any surface triangulation into any other of the same topological type. 5.3 Surface triangulation The sample distribution scheme just presented moves nodes... |

326 |
Introduction to Applied Mathematics
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- 1986
(Show Context)
Citation Context ... simply B 2 P) at each point on the boundary. This constraint cannot be directly enforced by freezing independent variables, as was done with point constraints, so a penalty[27] or Lagrange multiplier=-=[33]-=- technique should be used. 4.5 Minimizing the objective To minimizesE subject to the point constraints, we solve for the Q yielding rsE = 0. Rather than form H explicitly, the system collapse split sw... |

297 |
Differential Topology
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- 1974
(Show Context)
Citation Context ...The geometric fairing functionals of the next section will involve differentiation with respect to arc-length, 2 the Poincare-Hopf Index theorem on the existence of smooth vector-fields over manifolds=-=[15]-=-, which says, informally, that you cannot comb the hair on a ball without leaving a crown somewhere. q 1 q 2 q 3 q 4 q 5 aq 1 aq 2 aq 3 aq 4 aq 5 a( q 1 + q 2 + q 3 + q 4 + q 5 ) = 2p Figure 3: Surfac... |

242 | Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion
- Turk
- 1991
(Show Context)
Citation Context ...distribution We begin with the problem of maintaining a uniform sampling density over a surface. One approach that has been used successfully for both smooth and polyhedral surfaces is point repulsion=-=[29, 38,39,42]-=-: points move under the influence of mutual repulsive (or attractive) forces between points, constrained to act within the surface. In these schemes, a pair of points' influence on each other falls of... |

233 | Using particles to sample and control implicit surfaces
- Witkin, Heckbert
- 1994
(Show Context)
Citation Context ...distribution We begin with the problem of maintaining a uniform sampling density over a surface. One approach that has been used successfully for both smooth and polyhedral surfaces is point repulsion=-=[29, 38,39,42]-=-: points move under the influence of mutual repulsive (or attractive) forces between points, constrained to act within the surface. In these schemes, a pair of points' influence on each other falls of... |

205 |
Witkin A, Terzopoulos D. Snakes: Active Contour Models
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- 1987
(Show Context)
Citation Context ...nt sparsity without any additional effort on our part. 4.3 Curves Before taking up the constrained minimization of E, we briefly mention fairing for point-sampled curves. This is well-trodden ground (=-=[19]-=-), but to keep our presentation self-contained we point out that the geometric curve fairing objective E = Z 2 ds; (13) can be formulated analogously to the surface objective above. The shapes of spac... |

201 | Surface Modeling with Oriented Particle Systems
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- 1992
(Show Context)
Citation Context ...ace topologies interactively. The one approach to fair shape design that has allowed large-scale changes in shape and topology during sculpting is the oriented particle system of Szeliski and Tonnesin=-=[34]-=-. The drawback of this approach is that there is no explicit control over surface topology --- because there is no actual surface. A surface triangulation can be imposed on the particles strictly as a... |

194 |
Elementary Differential Geometry
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- 1966
(Show Context)
Citation Context ... the fairing functional (Section 4) a particularly simple form. 3.1 Neighborhood parameterizations Much of the theory of curves and surfaces is developed in terms of curves parameterized by arc-length=-=[25]-=-. Geometric quantities, such as curvature, have simple expressions with respect an arclength parameterization. The geometric fairing functionals of the next section will involve differentiation with r... |

188 | Mesh Generation and Optimal Triangulation
- Bern, Eppstein
- 1992
(Show Context)
Citation Context ... edges that must not be disturbed, such as those that are part of embedded control curves. A scheme that incorporates these source edges is referred to as a constrained Delaunay triangulation, or CDT =-=[2]-=-. It enjoys the same minimum-angle property as the DT (over all triangulations which include the source edges), and this leads directly to the flip algorithm for restoring a CDT given an initial trian... |

172 | Variational Surface Modeling
- Welch, Witkin
- 1992
(Show Context)
Citation Context ... 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 ... |

136 | A physically based approach to 2d shape blending
- Sederberg, Greenwood
- 1992
(Show Context)
Citation Context ... surfaces along a pair of boundaries, or skin two boundary curves with a single sheet, the nodes on the two curves must first be put into correspondence. In general, something like Sederberg's scheme =-=[31]-=- might be used to robustly determine this correspondence. A simpler (though by no means fail-safe) procedure is to iteratively refine the curve with fewer edges by splitting its longest edge until bot... |

119 | Direct Least-Squares Fitting of Algebraic Surfaces
- Pratt
- 1987
(Show Context)
Citation Context ...ties, there will be great advantage to having such derivatives be linear functions of the sample point positions. This rules out direct geometric constructions[21] as well as algebraic fitting methods=-=[28]-=-. Algebraic methods have the added drawback that they do not allow us to incorporate topological constraints like radial neighbor ordering into the fit. Instead, following a standard approach to const... |

115 |
Laplacian smoothing and Delaunay triangulation
- Field
- 1988
(Show Context)
Citation Context ... surface as if it were a (non-smooth) union of quadratic bowls, over which samples are sliding. As it happens, this method is closely related to a "mesh improvement " scheme called Laplacian=-= smoothing[11]-=-, so named because its fixed point is an approximate solution to Laplace's equation over the mesh. In fact, Laplace's equation is ubiquitous in computational mesh generation[36,37], arising naturally ... |

107 |
Deformable curve and surface finite-elements for free-form shape design
- Celniker, Gossard
- 1991
(Show Context)
Citation Context ... 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 ... |

56 |
Numerical Grid Generation. Foundations and Applications
- Thompson, Warsi, et al.
- 1001
(Show Context)
Citation Context ...led Laplacian smoothing[11], so named because its fixed point is an approximate solution to Laplace's equation over the mesh. In fact, Laplace's equation is ubiquitous in computational mesh generation=-=[36,37]-=-, arising naturally from a variational formulation of a uniform-density objective. We experimented with forming the Laplacian over a surface mesh, in terms of locally reconstructed neighborhoods[17], ... |

53 |
Scattered data interpolation and applications: A tutorial and survey
- Franke, Nielson
- 1991
(Show Context)
Citation Context ...dering for nodes beyond the immediate neighbors is not determined by the triangulation, so we do not search beyond the neighborhood to bring in additional samples as is common in least-squares schemes=-=[13]-=-). Worse, even though a node may have five neighbors they may be positioned parametrically so as to make the full biquadratic fit ill-conditioned. In this case, a reduced polynomial basis function is ... |

47 |
Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA
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- 1990
(Show Context)
Citation Context ...ack and forth between different sets of basis functions). Better would be to consistently use the same set of basis functions and optimize some auxiliary norm in the underdetermined case, as in Barth =-=[1]-=-. This requires an orthogonal decomposition of S T S, an added computational expense that degrades the overall interactivity of the modeler. 3.3 Curve reconstruction Smooth neighborhoods for PWL curve... |

43 |
An interpolation curve using a spline in tension
- Schweikert
- 1966
(Show Context)
Citation Context ... in which the topology remains fixed and surfaces do not stray far from their initial configurations. More drastic, nonuniform deformations are not handled well by the linearized thin plate functional=-=[30]-=- used to fair the piecewise smooth patches making up these surfaces. Further, no real consideration has been given to the problem of creating nontrivial smooth surface topologies interactively. The on... |

36 |
Finite difference methods for partial differential equations
- Forsythe, Wasow
- 1960
(Show Context)
Citation Context ...allow us to incorporate topological constraints like radial neighbor ordering into the fit. Instead, following a standard approach to constructing finite difference stencils over a computational field=-=[12]-=-, we will fit a truncated Taylor series expansion at a neighborhood around a point. This approach yields the desired linearity, but it comes at the expense of having to construct a separate bivariate ... |

23 | Difference formulas for the surface Laplacian on a triangulated surface
- Huiskamp
- 1991
(Show Context)
Citation Context ...36,37], arising naturally from a variational formulation of a uniform-density objective. We experimented with forming the Laplacian over a surface mesh, in terms of locally reconstructed neighborhoods=-=[17]-=-, and solving the resulting global error minimization for point positions. In practice, we found that this was more robust than the purely local scheme above, but was more expensive to compute, and in... |

23 |
Multi-level reconstruction of visual surfaces: Variational principles and finite element representations
- Terzopoulos
- 1982
(Show Context)
Citation Context ...rea form. Lott and Pullin [20] used this for surface fairing because of its relationship to the strain energy of a thin elastic plate. Unlike the more commonly used linearized thin plate approximation=-=[4,41,35,18]-=-, this formulation does not create shape artifacts related to an underlying fixed surface parameterization.sE is a geometric quantity whose definition is independent of parameterization. Recall from t... |

22 |
A translinite, visually continuous, triangular interpolant
- Nielson
- 1987
(Show Context)
Citation Context ...f the coarseness of their explicit approximations. If an explicit smooth surface is needed for any reason, it is a straightforward task to fit one to the triangulated surface in a post-processing step=-=[24]-=-. 1.3 Outline of this paper After a brief overview of our approach to variational sculpting and its underlying machinery, the bulk of the paper (sections 3--5) develops the mathematics and algorithms ... |

21 |
Constrained optimization in surface design
- Kallay
- 1993
(Show Context)
Citation Context ... 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 ... |

21 | A Survey of Parametric Scattered Data Fitting Using Triangular Interpolants. Curve and Surface
- Mann, Loop, et al.
- 1992
(Show Context)
Citation Context ...sitions that yield desired surface properties, there will be great advantage to having such derivatives be linear functions of the sample point positions. This rules out direct geometric constructions=-=[21]-=- as well as algebraic fitting methods[28]. Algebraic methods have the added drawback that they do not allow us to incorporate topological constraints like radial neighbor ordering into the fit. Instea... |

21 |
Constraint Methods for Neural Networks and Computer Graphics
- Platt
- 1989
(Show Context)
Citation Context ...aining s v (0; 0) (which is simply B 2 P) at each point on the boundary. This constraint cannot be directly enforced by freezing independent variables, as was done with point constraints, so a penalty=-=[27]-=- or Lagrange multiplier[33] technique should be used. 4.5 Minimizing the objective To minimizesE subject to the point constraints, we solve for the Q yielding rsE = 0. Rather than form H explicitly, t... |

20 |
Smooth interpolation of a mesh of curves
- Peters
- 1991
(Show Context)
Citation Context ...e. 7.2 Future Work Control nets: Though we formulate surface shape control using interpolated control curves, our scheme does not yet accommodate intersecting control curves. A compatibility condition=-=[26]-=- demands that when control curves meet at a point, they must all fit a common quadratic surface form; otherwise, no there can be no smooth interpolating surface in the neighborhood of the intersection... |

19 |
Guaranteed quality mesh generation for curved surfaces
- Chew
- 1993
(Show Context)
Citation Context ...combining a Laplacian neighborhood smoothing scheme and Delaunay triangulation (DT) tends to produce nice triangulations. For our meshes we will work with a generalization of the planar DT due to Chew=-=[6]. He gener-=-alizes the "empty circumcircle" characterization of the planar DT to one of empty circumspheres on a surface, and proves that it retains many of the desirable properties of the planar DT. Th... |

19 |
Topological design of sculptural surfaces
- Ferguson, Rockwood, et al.
- 1992
(Show Context)
Citation Context ...bution of this work is a scheme for interactively designing fair free-form shapes of arbitrary, mutable topology. Little work has appeared regarding topological design for freeform shapes (though see =-=[10]-=-). Our approach uses a triangulated mesh to represent a surface model's topology, and interactive modeling operations alter the mesh to change this topology in controlled ways. Geometric fairness func... |

17 |
Minimum Curvature Variation Curves, Networks, and Surfaces for Fair Free-Form Shape Design
- Moreton
- 1993
(Show Context)
Citation Context ...olling 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. I... |

13 |
A survey of dynamically-adaptive grids in numerical solution of partial dierential equations
- Thompson
- 1985
(Show Context)
Citation Context ...led Laplacian smoothing[11], so named because its fixed point is an approximate solution to Laplace's equation over the mesh. In fact, Laplace's equation is ubiquitous in computational mesh generation=-=[36,37]-=-, arising naturally from a variational formulation of a uniform-density objective. We experimented with forming the Laplacian over a surface mesh, in terms of locally reconstructed neighborhoods[17], ... |

12 |
Functional minimization for fair surface design
- Moreton, Séquin
- 1992
(Show Context)
Citation Context ...re configurations in which curvature minimization is undesirable as an objective function, as with narrow cylinders (which collapse). A minimum curvature variation functional, as in Moreton and Sequin=-=[22]-=-, would remedy this problem; but it is not clear how to compute the curvature derivatives given our local quadratic reconstructions, and we leave this as future work. 5 Surface sampling Taking the vie... |

9 |
Linear constraints for non-uniform Bspline surfaces
- Celniker, Welch
- 1992
(Show Context)
Citation Context |

5 |
Application of adaptive grids to fluid-flow problems with asymptotic solutions
- Rai, Anderson
- 1982
(Show Context)
Citation Context |

3 |
volume I
- Courant, Hilbert
- 1937
(Show Context)
Citation Context ...he understanding that operations will be interpreted as implicitly defining an ideal variational 1 We use "variational" in its mathematical sense as the solution of a problem in calculus of =-=variations[7]-=-. Unfortunately, this clashes with a different usage common in design literature. SIGGRAPH '94 Preprint 1 Not for distribution Figure 1: Making a Torus: 1. A closed curve is skinned to make a disc. 2.... |

3 |
Method for fairing b-spline surfaces. Computer-Aided Design 20
- LOTT, PULLIN
- 1988
(Show Context)
Citation Context ...the surface fairing objective function the integral of the squared principal curvatures over a smooth surface[25]: E = Z S ( 2 1 +s2 2 )dA; (7) where dA is the differential area form. Lott and Pullin =-=[20]-=- used this for surface fairing because of its relationship to the strain energy of a thin elastic plate. Unlike the more commonly used linearized thin plate approximation[4,41,35,18], this formulation... |

2 |
Pseudoedge: nonintersected parametric quilt modeling of multiply connected objects
- Bonner, Jakiela, et al.
- 1993
(Show Context)
Citation Context ...t arise. 6.6 Building structured models The composition of parameterized shapes via blending regions leads naturally to structured free-form models. As in Bonner, et al's work with tubular structures =-=[3]-=-, these shape control tools may be organized in a variety of ways through the use of deformation hierarchies (note that the deformations are not applied to the surface itself). In our system the resul... |

2 |
On the influence of parameterization in parametric interpolation
- Eppstein
- 1976
(Show Context)
Citation Context ... these points. Prior to the fit, we cannot assign these coordinates based on arclength because we have no curve. So instead, we will make a chord-length approximation to an arc-length parameterization=-=[9,8]-=-, in which the parametric interval between two samples is taken to be the 3D Euclidean distance between them. In constructing a neighborhood parameterization for surface fitting, it is common to proje... |