Simplicial isosurfacing in arbitrary dimension and codimension

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(Nⁿ) 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 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.