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The Distance Function from a Real Algebraic Variety

Giorgio Ottaviani, Luca Sodomaco

TL;DR

This work defines the Euclidean Distance polynomial EDpoly_{X,u}(t^2) for real varieties X in a Euclidean space with a quadratic form q, encapsulating the squared distance to X via its roots. It establishes a duality with the projective dual X^∨, namely EDpoly_{X,u}(t^2) = EDpoly_{X^∨,u}(q(u)−t^2), and analyzes how transversality to the isotropic quadric Q enforces a monic leading term and a structured lowest term. By leveraging conormal and Chern-Mather theory, the ED degree is expressed in terms of dual varieties and polar/Chern-Mather classes, with explicit formulas for hypersurfaces and matrix varieties. The paper provides concrete formulas and factorization patterns for essential cases, including the case of rank-constrained matrices and key hypersurface scenarios, linking distance geometry to dual geometry and offset hypersurfaces. These results offer deep geometric insight and concrete computational tools for distance problems in algebraic geometry and related applications.

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 $X\cup(X^\vee\cap Q)^\vee$.

The Distance Function from a Real Algebraic Variety

TL;DR

This work defines the Euclidean Distance polynomial EDpoly_{X,u}(t^2) for real varieties X in a Euclidean space with a quadratic form q, encapsulating the squared distance to X via its roots. It establishes a duality with the projective dual X^∨, namely EDpoly_{X,u}(t^2) = EDpoly_{X^∨,u}(q(u)−t^2), and analyzes how transversality to the isotropic quadric Q enforces a monic leading term and a structured lowest term. By leveraging conormal and Chern-Mather theory, the ED degree is expressed in terms of dual varieties and polar/Chern-Mather classes, with explicit formulas for hypersurfaces and matrix varieties. The paper provides concrete formulas and factorization patterns for essential cases, including the case of rank-constrained matrices and key hypersurface scenarios, linking distance geometry to dual geometry and offset hypersurfaces. These results offer deep geometric insight and concrete computational tools for distance problems in algebraic geometry and related applications.

Abstract

For any (real) algebraic variety in a Euclidean space endowed with a nondegenerate quadratic form , we introduce a polynomial which, for any , has among its roots the distance from to . The degree of is the {\em Euclidean Distance degree} of . We prove a duality property when is a projective variety, namely where is the dual variety of . When is transversal to the isotropic quadric , we prove that the ED polynomial of is monic and the zero locus of its lower term is .

Paper Structure

This paper contains 8 sections, 26 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition and first properties
  3. Projective duality corresponds to variable reflection in the ED polynomial
  4. The leading term of the ED polynomial
  5. The lowest term of the ED polynomial
  6. The ED degree in terms of dual varieties
  7. The ED polynomial of a hypersurface
  8. The case of matrices

Key Result

Theorem 1

Let $X\subset\mathbb{P}(V)$ be an irreducible variety and suppose that $X$ and $X^\vee$ are transversal to $Q$. Let $u\in V$ be a data point.

Figures (4)

  • Figure 1: A critical point $x\in X$ for the distance function $d_u$ on $X_{\mathrm{sm}}$. The squared norm $t^2=q(u-x)$ is a root of $\mathrm{EDpoly}_{X,u}$.
  • Figure 2: The dashed lines above form the quadric $Z_L$ when $L\subset\mathbb{C}^2$ is a point.
  • Figure 3: The example of the hyperbola $X$.
  • Figure 4: The cardioid $C$ with its ED dscriminant $\Sigma_C$ and the restriction to the affine plane of the ED discriminant $\Sigma_{\overline{C}}$ of its projectivization $\overline{C}$. A detail of $\Sigma_{\overline{C}}$ on the right.

Theorems & Definitions (64)

  • Theorem
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 54 more