Simplicial isosurfacing in arbitrary dimension and codimension

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Abstract

The level set method has been successfully used for moving interface problems. The final step of the method is to construct and visualize the isosurface of a discrete function φ : {0,...,N}n → ℝm. There have existed many practical isosurfacing algorithms when n = 3, m = 1 or n = 2, m = 1. Recently we have begun to see the development of isosurfacing algorithms for higher dimensions and codimensions. This paper introduces a unified theory and an efficient isosurfacing algorithm that works in arbitrary number of dimensions and codimensions. The isosurface Γ of a discrete function φ is defined as the isosurface of its simplicial interpolant ̂f : [0,N]n → ℝm. With this simplicial definition, Γ is geometrically a piecewise intersection of a simplex and m hyperplanes. Γ is constructed as the union of simplices. The construction costs O(Nn) with a uniform grid and O(Nn-m log(N)) with a dyadic grid in numerical space and time. When n = m + 1 or m + 2, Γ is projected down into ℝ3 and can be visualized. For surface visualizations, a simple formula is presented calculating the normal vector field of the projection of Γ into R3, which gives light shadings.

Original languageEnglish
Pages (from-to)295-310
Number of pages16
JournalJournal of Computational Physics
Volume190
Issue number1
DOIs
StatePublished - 1 Sep 2003

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