Lines, Circles, Planes and Spheres

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 $1+k\left(\genfrac{}{}{0ex}{}{n-k}{2}\right)-\left(\genfrac{}{}{0ex}{}{k}{2}\right)(\frac{n-k}{2})$. 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 $1+\left(\genfrac{}{}{0ex}{}{n-1}{3}\right)-t_3^{orchard}(n-1)$, 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.

37 pages

Cited by in corpus (2)

• Ordinary hyperspheres and spherical curves
• Ordinary planes, coplanar quadruples, and space quartics