TY - JOUR
T1 - Simplicial isosurfacing in arbitrary dimension and codimension
AU - Min, Chohong
N1 - Funding Information:
Research supported by ONR Grants N00014-02-1-0720, UCLA PY-2029 and NSF Grant DMS-0074735.
PY - 2003/9/1
Y1 - 2003/9/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0041355385&partnerID=8YFLogxK
U2 - 10.1016/S0021-9991(03)00275-4
DO - 10.1016/S0021-9991(03)00275-4
M3 - Article
AN - SCOPUS:0041355385
VL - 190
SP - 295
EP - 310
JO - Journal of Computational Physics
JF - Journal of Computational Physics
SN - 0021-9991
IS - 1
ER -