On the spherical clothoid

TL;DR

This paper explores the spherical clothoid, a nonlinear spline curve with linearly varying geodesic curvature, providing new mathematical representations and connections to special functions and polynomials.

Contribution

It introduces novel Cartesian coordinate functions and stereographic projection formulas for the spherical clothoid, along with new Humbert series results and generating functions.

Findings

Derived explicit Cartesian coordinate functions using confluent hypergeometric functions.

Presented stereographic projection formulas onto the complex plane.

Established new Humbert series and generating function results related to special polynomials.

Abstract

We revisit a nonlinear spline primitive for 3-space first studied by Even Mehlum. It is the spherical clothoid, the spherical curve with geodesic curvature a linear function of arc length. We present its Cartesian coordinate functions using confluent hypergeometric functions (the Kummer functions) and its stereographic projection onto the complex plane. New Humbert series results are also presented along with generating function formulas related to the associated Meixner-Pollaczek polynomials.