Integration over curves and surfaces defined by the closest point mapping

TL;DR

This paper introduces a novel method for integrating over curves and surfaces using closest point mappings, aligning volume integrals with surface or line integrals without explicit parameterizations, and extends it to 3D curves.

Contribution

The paper presents a new formulation for integration over manifolds via closest point mappings, compatible with level set methods, and extends it to 3D curves.

Findings

Exact volume integrals match desired surface or line integrals.

The formulation is effective with simple discretizations.

Provides geometric interpretation via Jacobian singular values.

Abstract

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wish to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation - initially derived to integrate over manifolds of codimension one - to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.