Finding an ordinary conic and an ordinary hyperplane

TL;DR

This paper extends algorithms for finding ordinary lines to compute ordinary conics and hyperplanes, providing simpler proofs and methods for these geometric structures in higher dimensions.

Contribution

It introduces simplified algorithms and proofs for computing ordinary conics and hyperplanes, generalizing existing line-based methods to higher-dimensional spaces.

Findings

Efficient algorithm for computing an ordinary conic in the plane.

Simpler proofs for the existence of ordinary conics and hyperplanes.

Extension of algorithms to higher dimensions for hyperplanes.

Abstract

Given a finite set of non-collinear points in the plane, there exists a line that passes through exactly two points. Such a line is called an ordinary line. An efficient algorithm for computing such a line was proposed by Mukhopadhyay et al. In this note we extend this result in two directions. We first show how to use this algorithm to compute an ordinary conic, that is, a conic passing through exactly five points, assuming that all the points do not lie on the same conic. Both our proofs of existence and the consequent algorithms are simpler than previous ones. We next show how to compute an ordinary hyperplane in three and higher dimensions.