Codimension-Two Objects

Abstract

Typically, level set methods are used to model codimension-one objects such as points in ℜ¹, curves in ℜ², and surfaces in ℜ³. Burchard, Cheng, Merriman, and Osher [22] extended level set technology to treat codimension-two objects using the intersection of the zero level sets of two level set functions. That is, instead of implicitly representing codimension-one geometry by the zero isocontour of a function φ, codimension-two geometry is represented as the intersection of the zero isocontour of a function φ₁ with the zero isocontour of another function φ_2. In one spatial dimension, zero isocontours are points, and their intersection is usually the empty set. In two spatial dimensions, zero isocontours are curves, and the intersections of curves tend to be points which are of codimension two. In three spatial dimensions, the zero isocontours are surfaces, and the intersections of these surfaces tend to be codimension two curves.