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17
FreeForm Shape Design Using Triangulated Surfaces
, 1994
"... We present an approach to modeling with truly mutable yet completely controllable freeform 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 ..."
Abstract

Cited by 153 (0 self)
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We present an approach to modeling with truly mutable yet completely controllable freeform 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 patchbased approaches. Triangulated point sets are used to approximate these smooth variational surfaces, bridging the gap between patchbased and particlebased representations. Automatic refinement, mesh smoothing, and retriangulation 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 ...
C¹ surface splines
 SIAM Journal of Numerical Analysis
, 1995
"... The construction of quadratic C¹
surfaces from Bspline control points is generalized to a wider class of control meshes capable of outlining arbitrary freeform surfaces in space. Irregular meshes with non quadrilateral cells and more or fewer than four cells meeting at a point are allowed so that ..."
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Cited by 33 (12 self)
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The construction of quadratic C¹
surfaces from Bspline control points is generalized to a wider class of control meshes capable of outlining arbitrary freeform surfaces in space. Irregular meshes with non quadrilateral cells and more or fewer than four cells meeting at a point are allowed so that arbitrary freeform surfaces with or without boundary can be modeled in the same conceptual frame work as tensorproduct Bsplines. That is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local, evaluates by averaging and obeys the convex hull property. For a regular region of the input mesh, the representation reduces to the standard quadratic spline. In general, any surface spline can be represented by BernsteinBezier patches of degree two and three. According to the user's choice, these patches can be polynomial or rational, threesided, foursided or a combination thereof.
Geometric Continuity
, 2001
"... This chapter covers geometric continuity with emphasis on a constructive definition for piecewise parametrized surfaces. The examples in Section 1 show the need for a notion of continuity different from the direct matching of Taylor expansions used to define the continuity of piecewise functions. Se ..."
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Cited by 12 (3 self)
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This chapter covers geometric continuity with emphasis on a constructive definition for piecewise parametrized surfaces. The examples in Section 1 show the need for a notion of continuity different from the direct matching of Taylor expansions used to define the continuity of piecewise functions. Section 2 defines geometric continuity for parametric curves, and for surfaces, first along edges, then around points, and finally for a whole complex of patches which is called a freeform surface spline. Here characterizes a relation between specific maps while continuity is a property of the resulting surface. The composition constraint on reparametrizations and the vertexenclosure constraints are highlighted. Section 3 covers alternative definitions based on geometric invariants, global and regional reparametrization and briefly discusses geometric continuity in the context of implicit representations and generalized subdivision. Section 4 explains the generic construction of freeform surface splines and points to some low degree constructions. The chapter closes with a listing of additional literature.
Surface Approximation Using Geometric Hermite Patches
, 1992
"... A highorderofapproximation surface patch is used to construct continuous, approximating surfaces. This patch, together with a relaxation of tangent plane continuity, is used to approximate offset surfaces, algebraic surfaces, and Spatches. ..."
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Cited by 11 (4 self)
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A highorderofapproximation surface patch is used to construct continuous, approximating surfaces. This patch, together with a relaxation of tangent plane continuity, is used to approximate offset surfaces, algebraic surfaces, and Spatches.
Smoothing Polyhedra Made Easy
, 1993
"... A mesh of points outlining a surface is polyhedral if all cells are either quadrilateral or planar. A mesh is vertexdegree bounded, if at most four cells meet at every vertex. This paper shows that if a mesh has both properties then simple averaging of its points yields the BernsteinB'ezier coeffi ..."
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Cited by 5 (3 self)
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A mesh of points outlining a surface is polyhedral if all cells are either quadrilateral or planar. A mesh is vertexdegree bounded, if at most four cells meet at every vertex. This paper shows that if a mesh has both properties then simple averaging of its points yields the BernsteinB'ezier coefficients of a smooth, at most cubic surface that consists of twice as many threesided polynomial pieces as there are interior edges in the mesh. Meshes with checker board structure, i.e. rectilinear meshes are a special case and result in a quadratic surface. Since any mesh and, in particular any wireframe of a polyhedron can be refined, by averaging, to a vertexdegree bounded polyhedral mesh this result allows reinterpreting a number of algorithms that construct smooth surfaces and advertises the corresponding averaging formulas as a basis for a wider class of algorithms. y Department of Computer Science, Purdue University, WLafayette IN 479071398 Supported by NSF grant CCR9211322 J P...
Fair Surface Reconstruction Using Quadratic Functionals
, 1995
"... An algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The first variant of the algorithm constructs a curve network consisting of cubic B'ezier curves meeting with tangent plane continuity at the vertices. This curve network is ..."
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Cited by 2 (0 self)
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An algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The first variant of the algorithm constructs a curve network consisting of cubic B'ezier curves meeting with tangent plane continuity at the vertices. This curve network is extended to a smooth surface by replacing each of the networks facets with a split patch consisting of three triangular B'ezier patches. The remaining degrees of freedom of the curve network and the split patches are determined by minimizing a quadratic functional. This optimization process works either for the curve network and the split patches separately or in one simultaneous step. The second variant of our algorithm is based on the construction of an optimized curve network with higher continuity. Examples demonstrate the quality of the different methods. 1 Introduction The reconstruction of a surface from a set of (a priori unorganized) points as well as the design of surfaces with a...
Polyhedral Modeling
"... Polyhedral meshes are used for visualization, computer graphics or geometric modeling purposes and result from many applications like isosurface extraction, surface reconstruction or CAD/CAM. The present paper introduces a method for constructing smooth surfaces from a triangulated polyhedral mesh ..."
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Cited by 2 (0 self)
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Polyhedral meshes are used for visualization, computer graphics or geometric modeling purposes and result from many applications like isosurface extraction, surface reconstruction or CAD/CAM. The present paper introduces a method for constructing smooth surfaces from a triangulated polyhedral mesh of arbitrary topology. It presents a new algorithm which generalizes and improves the triangle 4split method [7] in the crucial point of boundary curve network construction. This network is then filledin by a visual smooth surface from which an explicit closed form parametrization is given. Furthermore, the method becomes now completely local and can interpolate normal vector input at the mesh vertices. Keywords and phrases: triangular meshes, visual continuity, arbitrary topology, visualization. 1 INTRODUCTION Polyhedral meshes consisting of a collection of vertices, edges and triangular faces which describe an oriented 2manifold in IR 3 are dealt with in many applications. They resu...
From Degenerate Patches to Triangular and Trimmed Patches
 CURVES AND SURFACES
, 1997
"... CAD systems are usually based on a tensor product representation of free form surfaces. In this case, trimmed patches are used for modeling non rectangular zones. Trimmed patches provide a reasonable solution for the representation of general topologies, provided that the gap between equivalent trim ..."
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Cited by 1 (1 self)
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CAD systems are usually based on a tensor product representation of free form surfaces. In this case, trimmed patches are used for modeling non rectangular zones. Trimmed patches provide a reasonable solution for the representation of general topologies, provided that the gap between equivalent trimming curves in the euclidean space is small enough. Several commercial CAD systems, however, represent certain non rectangular surface regions through degenerate rectangular patches. Degenerate patches produce rendering artifacts and can lead to malfunctions in the subsequent geometric operations. In the present paper, two algorithms for converting degenerate tensorproduct patches into triangular and trimmed rectangular patches are presented. The algorithms are based on specific degree reduction algorithms for B'ezier curves. In both algorithms, the final surface approximates the initial one in a quadratic sense while inheriting its boundary curves. In the second one, " \Gamma G 1 cont...