The Distance Function from a Real Algebraic Variety

TL;DR

This paper introduces a polynomial that encodes the Euclidean distance from a point to a real algebraic variety, explores its duality properties, and characterizes its roots and degree in relation to the variety's geometry.

Contribution

It defines the ED polynomial for real algebraic varieties, proves a duality property for projective varieties, and characterizes the polynomial's roots and degree in special geometric cases.

Findings

The ED polynomial's roots include the distances from a point to the variety.

A duality relation links ED polynomials of a variety and its dual.

When the variety is transversal to the isotropic quadric, the ED polynomial is monic and its roots relate to the variety and its dual.

Abstract

For any (real) algebraic variety $X$ in a Euclidean space $V$ endowed with a nondegenerate quadratic form $q$, we introduce a polynomial $\mathrm{EDpoly}_{X,u}(t^2)$ which, for any $u\in V$, has among its roots the distance from $u$ to $X$. The degree of $\mathrm{EDpoly}_{X,u}$ is the {\em Euclidean Distance degree} of $X$. We prove a duality property when $X$ is a projective variety, namely $\mathrm{EDpoly}_{X,u}(t^2)=\mathrm{EDpoly}_{X^\vee,u}(q(u)-t^2)$ where $X^\vee$ is the dual variety of $X$. When $X$ is transversal to the isotropic quadric $Q$, we prove that the ED polynomial of $X$ is monic and the zero locus of its lower term is $X\cup(X^\vee\cap Q)^\vee$.