Abstract

The Jacobi set of two Morse functions defined on a common d-manifold is the set of critical points of the restrictions of one function to the level sets of the other function. Equivalently, it is the set of points where the gradients of the functions are parallel. For a generic pair of Morse functions, the Jacobi set is a smoothly embedded 1-manifold. We give a polynomial-time algorithm that computes the piecewise linear analog of the Jacobi set for functions specified at the vertices of a triangulation, and we generalize all results to more than two but at most d Morse functions.

Introduction

This paper is a mathematical and algorithmic study of multiple Morse functions, and in particular of their Jacobi sets. As we will see, this set is related to the Lagrange multiplier method in multi-variable calculus of which our algorithm may be viewed as a discrete analog.

Motivation. Natural phenomena are frequently modeled using continuous functions, and having two or more such functions defined on the same domain is a fairly common scenario in the sciences. Consider for example oceanography where researchers study the distribution of various attributes of water, with the goal to shed light on the ocean dynamics and gain insight into global climate changes [4]. One such attribute is temperature, another is salinity, an important indicator of water density. The temperature distribution is often studied within a layer of constant salinity, because water tends to mix along but not between these layers.