Signed Distance Functions

Abstract

In the last chapter we defined implicit functions with φ(x↦) ≤ 0 in the interior region Ω^-, φ((x↦) > 0 in the exterior region Ω⁺, and φ((x↦) = 0 on the boundary ∂Ω. Little was said about φ otherwise, except that smoothness is a desirable property especially in sampling the function or using numerical approximations. In this chapter we discuss signed distance functions, which are a subset of the implicit functions defined in the last chapter. We define signed distance functions to be positive on the exterior, negative on the interior, and zero on the boundary. An extra condition of |∇φ(x↦)| = 1 is imposed on a signed distance function.