Metric Problems for Quadrics in Multidimensional Space

TL;DR

This paper develops a method using Elimination Theory to compute the squared distance between quadrics in multidimensional space, extending it to parameter-dependent surfaces.

Contribution

It introduces a novel approach for deriving a univariate polynomial whose roots correspond to squared distances between quadrics, including parameter-dependent cases.

Findings

Derived a polynomial for squared distances between quadrics

Extended method to parameter-dependent surfaces

Provides a systematic algebraic approach

Abstract

Given the equations of the first and the second order surfaces in multidimensional space, our goal is to construct a univariate polynomial one of the zeros of which coincides with the square of the distance between these surfaces. To achieve this goal we employ Elimination Theory methods. The proposed approach is also extended for the case of parameter dependent surfaces.