Euclidean distance discriminants and Morse attractors

TL;DR

This paper analyzes the Euclidean distance function on complex plane curves, decomposing its discriminant into components responsible for different Morse point behaviors, including a unique phenomenon at infinity.

Contribution

It introduces a decomposition of the Euclidean distance discriminant into components, revealing new behaviors specific to complex plane curves and formulas for Morse singularity counts.

Findings

Decomposition of ED discriminant into three components.

Identification of an atypical discriminant at infinity.

Formulas for Morse singularity counts near discriminant components.

Abstract

Our study concerns the Euclidean distance function in case of complex plane curves. We decompose the ED discriminant into components which are responsible for three types of behavior of the Morse points. Besides the traditional focal component, which is non--linear; the other components are lines. In particular we shed light on the ``atypical discriminant'' which is due to the loss of Morse critical points at isotropic points at infinity. This phenomenon is specific for the complex setting. We find formulas for the number of Morse singularities which abut to the corresponding type of attractors when moving the centre of the distance function toward a point of the discriminant.