We present a general method for automatic reconstruction of accurate, concise, piecewise smooth surface models from scattered range data. The method can be used in a variety of applications such as reverse engineering — the automatic generation of CAD models from physical objects. Novel aspects of the method are its ability to model surfaces of arbitrary topological type and to recover sharp features such as creases and corners. The method has proven to be effective, as demonstrated by a number of examples using both simulated and real data. A key ingredient in the method, and a principal contribution of this paper, is the introduction of a new class of piecewise smooth surface representations based on subdivision. These surfaces have a number of properties that make them ideal for use in surface reconstruction: they are simple to implement, they can model sharp features concisely, and they can be fit to scattered range data using an unconstrained optimization procedure.

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 ...

quadratically varying normals). To improve the visual quality of existing triangle-based art in realtime entertainment, such as computer games, we propose replacing flat triangles with curved patches and higher-order normal variation. At the hardware level, based only on the three vertices and three vertex normals of a given flat triangle, we substitute the geometry of a three-sided cubic Bézier patch for the triangle’s flat geometry, and a quadratically varying normal for Gouraud shading. These curved point-normal triangles, or PN triangles, require minimal or no change to existing authoring tools and hardware designs while providing a smoother, though not necessarily everywhere tangent continuous, silhouette and more organic shapes. CR Categories: I.3.5 [surface representation, splines]: I.3.6— graphics data structures

This paper has been published as a chapter in "Curve and Surface Design", H. Hagen, (ed), SIAM, 1992 Some of the figures from that paper are missing from this version, as are all of the black-and-white photographs. There are currently a number of methods for solving variants of the following problem: Given a triangulated polyhedron P in three space with or without boundary, construct a smooth surface that interpolates the vertices of P. In general, while the methods satisfy the continuity and interpolation requirements of the problem, they often fail to produce pleasing shapes. The purpose of this paper is to present a unifying survey of the published methods, to identify causes of shape defects, and to offer suggestions for improving the aesthetic quality of the interpolants. 8.1 Introduction.

A high-order-of-approximation 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 S-patches.

The problem of interpolating scattered 3D data by a geometrically smooth surface is considered. A completely local method is proposed, based on employing degenerate triangular Bernstein-Bézier patches. An analysis of these patches is given and some numerical experiments with quartic and quintic patches are presented.

There are currently a number of methods for solving variants of the following problem: Given a triangulated polyhedron P in three-space with or without boundary, construct a smooth surface that interpolates the vertices of P. Problems of this variety arise in numerous areas of application such as medical imaging, scattered data fitting, and geometric modeling. In general, while the techniques satisfy the continuity and interpolation requirements of the problem, they often fail to produce pleasing shapes. Our interest in studying this problem has necessitated the construction of a flexible software testbed that allows rapid implementation and testing of new surface fitting methods and analysis techniques. The testbed is written entirely in the C programming language and is highly portable. Other relevant features of the testbed are discussed, and recommendations for improving the shape characteristics of several interpolation methods are given. 1 MOTIVATION There is much research to be...

This survey presents an overview to various types of continuity of curves and surfaces, in particular parametric (C n ), visual or geometric (V n , G n ), Frenet frame (F n ), and tangent surface continuity (T n ), and discusses the relation with curve and surface modeling, visibility of (dis)continuities, and graphics rendering algorithms. It is the purpose of this paper to provide an overview of types of continuity, and to put many terms and definitions on a common footing in order to give an understanding of the subject. 1991 Mathematics Subject Classification: 41A15, 65D07, 68U05 1991 Computing Reviews Classification: I.3.5 [Computer Graphics] Computational geometry and object modeling. I.3.7 [Computer Graphics] Three-Dimensional Graphics and Realism. Key Words and Phrases: Curves, Surfaces, Continuity, Shading, Modeling. 1

A piecewise quintic G¹ spline surface interpolating the vertices of a triangular surface mesh of arbitrary topological type is presented. The surface has an explicit triangular B'ezier representation, is affine invariant and has local support. The twist compatibility problem which arises when joining an even number of polynomial patches G¹ continuously around a common vertex is solved by constructing C²-consistent boundary curves. Piecewise C¹ boundary curves and a regular 4-split of the domain triangle make shape parameters available for controlling locally the boundary curves. A small number of free inner control points can be chosen for some additional local shape effects.

Sophia-Antipolis, France, during the summer of 2007. We are grateful to Nicholas Ayache and the members of the ASCLEPIOS group for their hospitality. Some people have read incomplete versions of the manuscript, pointed out several mistakes, and given useful suggestions. In particular, we thank Dimas Martínez, Luis Gustavo Nonato, and Luiz Velho for that. All of them are currently working with us on some extensions of the results described in this manuscript. Almost all triangle meshes used in the experiments described in this manuscript belong to the repository of 3D digital models of the AIM@SHAPE project. We also relied upon several open source and freely available softwares, such as MeshLab, Geomview, SurfRemesh and Blender, for preprocessing the triangle meshes we used. Luiz Velho also provided us with a software for mesh simplification based on his Four-Face Clusters algorithm. Marcelo Siqueira would like to thank CNPq for partially supporting his research (Grant 475703/2006-