Applying inversion to construct rational spiral curves

TL;DR

This paper introduces a method to construct rational spiral curves through inversion of parabola arcs, enabling flexible boundary condition matching including tangents and curvatures, even with inflection points.

Contribution

It presents a novel approach to generate rational spiral curves of 4th order using inversion, expanding the tools for curve design in geometric modeling.

Findings

The method successfully constructs spiral curves matching specified boundary conditions.

It allows for the inclusion of inflection points in the spiral curves.

The resulting curves are rational of 4th order, suitable for practical applications.

Abstract

A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including those, assuming inflection.