Ten Points on a Cubic

TL;DR

This paper introduces a straightedge-based method to verify if ten points lie on a plane cubic curve, extending Pascal's classical conic test to more complex algebraic curves.

Contribution

We develop a novel geometric technique using straightedge constructions to determine if ten points are on a cubic curve, generalizing Pascal's classical conic test.

Findings

Method successfully verifies point-on-cubic conditions

Extends classical conic testing to cubic curves

Provides a geometric approach for algebraic curve analysis

Abstract

The 16-year old Blaise Pascal found a way to determine if 6 points lie on a conic using a straightedge. Nearly 400 years later, we develop a method that uses a straightedge to check whether 10 points lie on a plane cubic curve.