Recognizing projections of rational curves

TL;DR

This paper presents an algorithm to determine if one rational space curve can be projected onto a planar rational curve via perspective or parallel projection, and identifies the projection parameters.

Contribution

It introduces a symbolic and approximate algorithm for recognizing projections of rational curves, including computing the projection parameters.

Findings

Algorithm successfully identifies projection existence

Provides explicit projection parameters in affirmative cases

Includes both symbolic and approximate methods

Abstract

Given two rational, properly parametrized space curves ${\mathcal C}_1$ and ${\mathcal C}_2$, where $\CCC_2$ is contained in some plane $\Pi$, we provide an algorithm to check whether or not there exist perspective or parallel projections mapping $\CCC_1$ onto $\CCC_2$, i.e. to recognize $\CCC_2$ as the projection of $\CCC_1$. In the affirmative case, the algorithm provides the eye point(s) of the perspective transformation(s), or the direction(s) of the parallel projection(s). The problem is mainly discussed from a symbolic point of view, but an approximate algorithm is also included.