Detecting Similarity of Rational Plane Curves

TL;DR

This paper introduces a deterministic algorithm to identify if two rational plane curves are related by a similarity transformation, also finding all such transformations and symmetries, with implementation in Sage for moderate degrees.

Contribution

It presents a new, complete algorithm for detecting similarities between rational plane curves, including symmetries, with theoretical analysis and practical implementation.

Findings

Algorithm successfully detects similarities and symmetries.

Implemented in Sage and tested on curves of moderate degree.

Provides a complete theoretical framework for the method.

Abstract

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.