## 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} → ℝ^{m}. 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} → ℝ^{m}. 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(N^{n}) with a uniform grid and O(N^{n-m} log(N)) with a dyadic grid in numerical space and time. When n = m + 1 or m + 2, Γ is projected down into ℝ^{3} and can be visualized. For surface visualizations, a simple formula is presented calculating the normal vector field of the projection of Γ into R^{3}, which gives light shadings.

Original language | English |
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Pages (from-to) | 295-310 |

Number of pages | 16 |

Journal | Journal of Computational Physics |

Volume | 190 |

Issue number | 1 |

DOIs | |

State | Published - 1 Sep 2003 |

### Bibliographical note

Funding Information:Research supported by ONR Grants N00014-02-1-0720, UCLA PY-2029 and NSF Grant DMS-0074735.