Lines, Circles, Planes and Spheres
George B. Purdy, Justin W. Smith

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