Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces

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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.

Original languageEnglish (US)
Pages (from-to)527-545
Number of pages19
JournalVisual Computer
Issue number10
StatePublished - Jan 1 1996
Externally publishedYes


  • Distance
  • Implicit surface
  • Lipschitz condition
  • Ray tracing
  • Solid modeling

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computer Graphics and Computer-Aided Design


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