A Note on Robust Biarc Computation

TL;DR

This paper introduces a robust algebraic algorithm for computing biarcs, which are $G^1$ continuous curves made of two circular arcs, overcoming limitations of geometric methods especially near singular configurations.

Contribution

The paper presents an algebraic approach to biarc computation that reliably handles singular configurations using pseudoinverse matrices, improving robustness over existing geometric algorithms.

Findings

The proposed algorithm always finds the correct biarc solution near singular configurations.

It outperforms Matlab's exttt{rscvn} routine in accuracy and reliability.

The method smoothly depends on parameters, facilitating integration into complex curve fitting algorithms.

Abstract

A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical configurations. The proposed algorithm uses an algebraic construction which is reduced to the solution of a single $2$ by $2$ linear system. Singular angles configurations are treated smoothly by using the pseudoinverse matrix when solving the linear system. The proposed algorithm is compared with the Matlab's routine \texttt{rscvn} that solves geometrically the same problem. Numerical experiments show that Matlab's routine sometimes fails near singular configurations and does not select the correct solution for large angles, whereas the proposed algorithm always returns the correct solution. The proposed solution smoothly depends on the geometrical parameters so that it can be easily included in more complex algorithms like splines of biarcs or least squares data fitting.