Lines, Circles, Planes and Spheres

TL;DR

This paper establishes new lower bounds on the number of planes, spheres, lines, and circles determined by large point sets in three-dimensional space, extending classical combinatorial geometry results.

Contribution

It introduces novel lower bounds for the number of geometric objects determined by point sets in , generalizing and improving upon existing results, including the Orchard Problem.

Findings

Lower bound on the number of planes determined by points in .

Lower bound on the number of spheres determined by points in .

New bounds for lines and circles determined by point sets.

Abstract

Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k \binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})$. For similar conditions and sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in \cite{Ell67}) we also show that the number of spheres determined by $n$ points is at least $1+\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best possible under its hypothesis. (By $t_3^{orchard}(n)$, we are denoting the maximum number of three-point lines attainable by a configuration of $n$ points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles.