July 29, 2004

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Publications by

• Using the CW-Complex to Represent the Topological Structure of Implicit Surfaces and Solids, 1999.
• Computational Topology for Shape Modeling, 1999.
• Critical Points of Polynomial Metaballs, 1998.
• Morse Theory for Implicit Surface Modeling, 1997-98.
• Morse Theory for Computer Graphics, 1997.
• Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces, 1996.

BibTeX references.

Web link: http://citeseer.nj.nec.com/287801.html

We investigate the CW-complex as a data structure for visualizing and controlling the topology of implicit surfaces. Previous methods for contolling the blending of implicit surfaces redefined the contribution of a metaball or unioned blended components. Morse theory provides new insight into the topology of the surface a function implicitly defines by studying the critical points of the function. These critical points are organized by a separatrix structure into a CW-complex. This CW-complex forms a topological skeleton of the object, indicating connectedness and the possibility of connectedness at various locations in the surface model. Definitions, algorithms and applications for the CW-complex of an implicit surface and the solid it bounds are given as a preliminary step toward direct control of the topology of an implicit surface.

Web link : http://citeseer.nj.nec.com/282113.html

This paper expands the role of the new field of computational topology by surveying methods for incorporating connectedness in shape modeling. Two geometric representations in particular, recurrent models and implicit surfaces, can (often unpredictably) become connected or disconnected based on typical changes in modeling parameters. Two methodologies for controlling connectedness are identified: connectedness loci and Morse theory. The survey concludes by identifying several open problems remaining in shape modeling for computational topology to solve.

Morse theory describes the relationship between a function's critical points and the homotopy type of the function's domain. The theorems of Morse theory were developed specifically for functions on a manifold. This work adapts these theorems for use with implicit surfaces. The result is a theoretical basis for the determination of the global topology of an implicit surface, and in the process, the steps of a fundamental proof in Morse theory are visualized.

Morse theory is an area of algebraic topology concerned with the affect of critical points on a set's topology. This report identifies the key theorems from Morse theory and presents them in a way that may be easier to understand for the computer graphics researcher. As the theorems of Morse theory do not directly apply to the common computer graphics representations, several corollaries are stated and proven that map the Morse theory results to implicit surfaces and extensions to other geometric representations are described.

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