Polar varieties revisited

TL;DR

This paper revisits classical and reciprocal polar varieties, exploring their geometric properties and relationships with Euclidean distance degree and focal loci, especially in the context of singular varieties.

Contribution

It provides a comprehensive review of polar varieties and introduces their connection to Euclidean geometry and singularity theory.

Findings

Recalls definitions of classical and reciprocal polar varieties.

Establishes links between polar varieties and Euclidean geometry.

Expresses Euclidean distance degree and focal loci degree via polar varieties.

Abstract

We recall the definition of classical polar varieties, as well as those of affine and projective reciprocal polar varieties. The latter are defined with respect to a non-degenerate quadric, which gives us a notion of orthogonality. In particular we relate the reciprocal polar varieties to the "Euclidean geometry" in projective space. The Euclidean distance degree and the degree of the focal loci can be expressed in terms of the ranks, i.e., the degrees of the classical polar varieties, and hence these characters can be found also for singular varieties, when one can express the ranks in terms of the singularities.