Partial Fraction Decomposition for Rational Pythagorean Hodograph Curves

TL;DR

This paper explores the structure of rational Pythagorean hodograph curves by analyzing vector spaces defined by fixed denominator polynomials, introducing a partial fraction-like decomposition method, and discussing implications for interpolation.

Contribution

It introduces a canonical basis construction for these vector spaces and demonstrates a partial fraction decomposition analogy for rational curves.

Findings

Canonical bases for fixed denominator subspaces are constructed.

Any rational Pythagorean hodograph curve can be expressed as a sum of curves with single roots.

Insights into the structure and applications of these curve spaces are provided.

Abstract

All rational parametric curves with prescribed polynomial tangent direction form a vector space. Via tangent directions with rational norm, this includes the important case of rational Pythagorean hodograph curves. We study vector subspaces defined by fixing the denominator polynomial and describe the construction of canonical bases for them. We also show (as an analogy to the fraction decomposition of rational functions) that any element of the vector space can be obtained as a finite sum of curves with single roots at the denominator. Our results give insight into the structure of these spaces, clarify the role of their polynomial and truly rational (non-polynomial) curves, and suggest applications to interpolation problems.