When do Ten Points Lie on a Quadric Surface?

TL;DR

This paper provides a synthetic geometric solution to the Bruxelles Problem, determining when ten points in 3D space lie on a quadric surface using classical and computational algebraic tools.

Contribution

It introduces a novel synthetic construction and extends classical geometric principles to solve a longstanding problem with modern algebraic methods.

Findings

Constructs four points coplanar iff original points lie on a quadric surface.

Uses an extension of the Area Principle, bracket polynomials, and Grassmann-Cayley algebra.

Identifies a basis for quadrics passing through six points using Macaulay2 insights.

Abstract

A solution is provided to the Bruxelles Problem, a geometric decision problem originally posed in 1825, that asks for a synthetic construction to determine when ten points in 3-space lie on a quadric surface, a surface given by the vanishing of a degree-2 polynomial. The solution constructs four new points that are coplanar precisely when the ten original points lie on a quadric surface. The solution uses only lines constructed through two known points, planes constructed through three known points, and intersections of these objects. The tools involved include an extension of the Area Principle to three-dimensional space, bracket polynomials and the Grassmann-Cayley algebra, and von Staudt's results on geometric arithmetic. Many special cases are treated directly, leading to the generic case, where three pairs of the points generate skew lines and the remaining four points are in general position. A key step in the generic case involves finding a nice basis for the quadrics that pass through six of the ten points, which uses insights derived from Macaulay2, a computational algebra package not available in the nineteenth century.