Simplicial isosurfacing in arbitrary dimension and codimension

• Authors: Chohong Min
• Venue: Journal of Computational Physics
• Year: 2003

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 φ:[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.