Construction of a spiral with given boundary conditions by inversion of the involute of a circle

TL;DR

This paper presents a method for constructing a spiral curve with specified boundary tangents and curvatures by using involutes of a circle and linear-fractional maps, simplifying the process.

Contribution

The paper introduces a novel approach using involutes of a circle and linear-fractional transformations to construct boundary-conditioned spirals.

Findings

The method effectively constructs spirals with given boundary conditions.

Using involutes of a circle simplifies the construction process.

Linear-fractional maps facilitate solving the boundary problem.

Abstract

To construct a curve with a monotonic curvature (spiral), and given tangents and curvatures at the ends, the author proposed the following method. From given boundary conditions, the values of two inverse invariants are determined. Then, on some base spiral (initially, a logarithmic spiral was chosen), an arc with the same invariant values is sought for. A linear-fractional map of the found arc solves the problem. It seems that choosing the involute of a circle as the base spiral yields the simplest solution, which we present here.