Three Paths to Rational Curves with Rational Arc Length

TL;DR

This paper introduces three novel methods for constructing all spatial rational curves with rational arc length functions, expanding the understanding and tools for designing such curves in geometric modeling.

Contribution

It presents three distinct approaches for constructing rational curves with rational arc length functions, extending existing methods and providing new theoretical insights.

Findings

Three methods successfully construct all such curves.

Extension of dual approach from planar to spatial curves.

New proof based on quaternion polynomial factorization.

Abstract

We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first method adapts a recent approach of (Kalkan et al. 2022) to rational PH curves and requires solving a modestly sized system of linear equations. The second constructs the curve by imposing zero-residue conditions, thus extending ideas of previous papers by (Farouki and Sakkalis 2019) and the authors themselves (Schr\"ocker and \v{S}\'ir 2023). The third method generalizes the dual approach of (Pottmann 1995) from planar to spatial curves. The three methods share the same quaternion based representation in which not only the PH curve but also its arc length function are compactly expressed. We also present a new proof based on the quaternion polynomial factorization theory of the well known characterization of the Pythagorean quadruples.