Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves

TL;DR

This paper introduces a novel method for constructing rational spatial Pythagorean Hodograph curves using rational framing motions, revealing conditions for truly rational solutions and providing comprehensive characterizations and examples.

Contribution

It presents a new approach based on rational framing motions for constructing rational PH curves, including a full characterization of cases with non-polynomial solutions.

Findings

Generic data yields polynomial PH curves

Complete characterization of rational (non-polynomial) solutions

Method simplifies construction of rational PH curves

Abstract

We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing motion is arbitrary, the translation part is determined be a modestly sized and nicely structured system of linear equations. Rather surprisingly, generic input data will only result in polynomial PH curves. We provide a complete characterization of all cases that admit truly rational (non-polynomial) solutions. Examples illustrate our ideas and relate them to existing literature.