Three Paths to Rational Curves with Rational Arc Length

TL;DR

The paper tackles the problem of constructing all spatial rational curves with rational arc length by presenting three universal methods that share a quaternion-based representation of the hodograph and arc length. The first method solves a linear system for a rational hodograph via $\alpha(t)\dot{\mathbf B}(t)-\dot{\alpha}(t)\mathbf B(t)=\mu(t)\mathcal A(t)\mathbf i\mathcal A^*(t)$, the second imposes zero-residue conditions on the rational arc-length integrand (augmented with $\mathcal F(t)=\mathcal A(t)(1+\mathbf i)\mathcal A^*(t)$ for the arc length), and the third, a geometric dual approach, constructs the curve as the envelope of osculating hyperplanes yielding a rational curve of constant slope $1$ in $\mathbb H$. A new quaternion-factorization-based proof of the PH quadruple characterization is provided, and the authors compare the methods in terms of universality, degree control, and applicability to interpolation and design. Together, these results offer a unified, versatile toolkit for generating all spatial rational PH curves with rational arc length and shed light on the structural role of quaternion polynomials in these constructions.

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 (Schröcker and Šír 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.

Figures (1)

• Figure 1: Hermite interpolation by a curve with rational arc length.

Theorems & Definitions (13)

• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Corollary 2.4
• Theorem 3.1
• Theorem 3.4
• proof : Proof of Theorem \ref{['th:linear-system']}
• Theorem 4.1
• proof