Jan. 14, 2001

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Publications by John C. Hart et al., School of EECS, Washington State University, Pullman, WA, hart@eecs.wsu.edu :

• 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.

For more on this topic by Hart et al., click here.

Sphere tracing is a new technique for rendering implicit surfaces that uses geometric distance. Sphere tracing marches along the ray toward its first intersection in steps guaranteed not to penetrate the implicit surface. It is particularly adept at rendering pathological surfaces. Creased and rough implicit surfaces are defined by functions with discontinuous or undefined derivatives. Sphere tracing requires only a bound on the magnitude of the derivative, robustly avoiding problems where the derivative jumps or vanishes. It is an efficient direct visualization system for the design and investigation of new implicit models. Sphere tracing efficiently approximates cone tracing, supporting symbolic-prefiltered antialiasing. Signed distance functions for a variety of primitives and operations are derived.

Keywords: Distance · Implicit surface · Lipschitz condition · Ray tracing · Solid modeling

Summary

1. Introduction

1.1. Previous work

1.2 Overview

2. Sphere Tracing

Sphere tracing capitalizes on functions that return the distance to their implicit surfaces.

2.1 Distance Surfaces

Consider functions that measure or bound the geometric distance to their implict surfaces: such functions implicitly define distance surfaces.

2.2 Ray Intersection

2.3 Analysis

2.4 Lipschitz Methods vs. Interval Analysis

2.5 Constructive Solid Geometry

2.6 Enhancements

2.6.1 Image Coherence

Includes distance transforms.

2.6.2 Bounding Volumes

2.6.3 The Triangle Inequality

2.6.4 Octree Partitioning

2.6.5 Convexity

3. Antialiasing

4. Results

4.1 Implementation

4.3 Analysis

5. Conclusion

5.1 Further Research

Demand for more efficient geometric distance algorithms will increase.

A: Distance to Natural Quadrics & Torus

B: Distance to Superquadrics

C: Distance to Offset Surfaces

D: Distance to Blended Objects

D.1 Soft Metablobbies

D.2 Superelliptic Blends

E: Distance to Transformed Objects

F: Distance to Hypertextures

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