Interactive Visualization of Implicit Surfaces with Singularities

This paper presents work on two methods for interactive visualization of implicit surfaces: physically‐based sampling using particle systems and polygonization followed by physically‐based mesh improvement which explicitly makes use of the surface‐defining equation. While most previous work applied to bounded manifolds without singularities and without boundary (topological spheres) we broaden the scope of the methods to include surfaces with such features, in particular cusp points and surface self‐intersections. These aspects are not (yet) essential for computer graphics modelling with implicit surfaces but they naturally occur in simulations of interest in mathematical visualization. In this paper we use the Kummer family of algebraic surfaces as an example.

[1]  D. A. Field.  Laplacian smoothing and Delaunay triangulations , 1988 .

[2]  Jules Bloomenthal,et al.  Polygonization of non-manifold implicit surfaces , 1995, SIGGRAPH.

[3]  Dietmar Saupe,et al.  Animation of Algebraic Surfaces , 1995, VisMath.

[4]  Thomas W. Sederberg,et al.  Scan line display of algebraic surfaces , 1989, SIGGRAPH.

[5]  William E. Lorensen,et al.  Decimation of triangle meshes , 1992, SIGGRAPH.

[6]  Jules Bloomenthal,et al.  Polygonization of implicit surfaces , 1988, Comput. Aided Geom. Des..

[7]  Luiz Velho,et al.  Physically-based methods for polygonization of implicit surfaces , 1992 .

[8]  Paul S. Heckbert,et al.  Using particles to sample and control implicit surfaces , 1994, SIGGRAPH.

[9]  Greg Turk,et al.  Re-tiling polygonal surfaces , 1992, SIGGRAPH.

[10]  Mark Hall,et al.  Adaptive polygonalization of implicitly defined surfaces , 1990, IEEE Computer Graphics and Applications.

[11]  Marilyn E. Noz,et al.  Constructing topologically connected surfaces for the comprehensive analysis of 3-D medical structures , 1991, Medical Imaging.

[12]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[13]  HallMark,et al.  Adaptive Polygonalization of Implicitly Defined Surfaces , 1990 .

[14]  Jean-Christophe Lombardo,et al.  Oriented particles: A tool for shape memory objects modelling , 1995 .

[15]  Kenji Shimada,et al.  Bubble mesh: automated triangular meshing of non-manifold geometry by sphere packing , 1995, SMA '95.

[16]  Hartmut Jürgens,et al.  Continuation Methods for Approximating Isovalued Complex Surfaces , 1991, Eurographics.

[17]  Richard Szeliski,et al.  Surface modeling with oriented particle systems , 1992, SIGGRAPH.

[18]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[19]  Tony DeRose,et al.  Mesh optimization , 1993, SIGGRAPH.

[20]  Greg Turk,et al.  Generating textures on arbitrary surfaces using reaction-diffusion , 1991, SIGGRAPH.

[21]  Paul Ning,et al.  An evaluation of implicit surface tilers , 1993, IEEE Computer Graphics and Applications.

[22]  Pat Hanrahan,et al.  Ray tracing algebraic surfaces , 1983, SIGGRAPH.

[23]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .