Euclidean 1-center of a set of static and mobile points

TL;DR

This paper derives the exact algebraic form of the Euclidean 1-center function for static and mobile points in multi-dimensional space, providing a computational algorithm for its parametric equation.

Contribution

It introduces a method to compute the algebraic parametric equation of the Euclidean 1-center function for static and mobile points with rational motion.

Findings

The Euclidean 1-center function is piecewise differentiable.

An algorithm to compute the parametric equation of the 1-center function is proposed.

The exact algebraic form of the 1-center function is derived.

Abstract

In this paper, we consider the problem of computing the algebraic parametric equation of the Euclidean 1-center function in $\mathbb{R}^d$, $d \geq 2$, for a system of $n$ static points and $m$ mobile points having motion defined by rational parametric functions. We have shown that the corresponding Euclidean 1-center function is a piecewise differentiable function and have derived its exact parametric algebraic equation. If the positions of the static points and the rational parametric equations of the motion of the mobile points are given, we have proposed an algorithm that computes the parametric equation of the Euclidean 1-center function.