Similarity detection of rational space curves

TL;DR

This paper presents an algorithm to determine if two rational space curves are related by a similarity transformation, utilizing curvature and torsion relationships, with practical implementation details and experiments.

Contribution

It introduces a novel algorithm that leverages curvature and torsion to detect similarities between rational space curves, including special cases like helical curves.

Findings

Algorithm successfully distinguishes similar curves using algebraic techniques.

Implementation in Maple 18 demonstrates practicality and effectiveness.

Special handling for helical curves improves accuracy.

Abstract

We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Gr\"obner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.