Justin W. Smith

Lines, Circles, Planes and Spheres

Let S be a set of n points in ℝ3, 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

Bichromatic and Equichromatic Lines in ℂ 2 and ℝ 2

1+k\binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})

A Multi-Stage CUDA Kernel for Floyd-Warshall

. For similar conditions and sufficiently large n, (inspired by the work of P.D.T.A. Elliott in Acta Math. Sci. Hung. 18:181–188, 1967) we also show that the number of spheres determined by n points is at least

A Bichromatic Incidence Bound and an Application

1+\binom{n-1}{3}-t_{3}^{\mathrm{orchard}}(n-1)

Some Results Related to a Conjecture of Dirac's

, and this bound is best possible under its hypothesis. (By

Problems and Results in Discrete and Computational Geometry

t_{3}^{\mathrm{orchard}}(n)

Collinearities in Kinetic Point Sets

, 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.

Points and Lines in the Plane

Let G and R each be a finite set of green and red points, respectively, such that |G|=n, |R|=n−k, G∩R=∅, and the points of G∪R are not all collinear. Let t be the total number of lines determined by G∪R. The number of equichromatic lines (a subset of bichromatic) is at least (t+2n+3−k(k+1))/4. A slightly weaker lower bound exists for bichromatic lines determined by points in ℂ2. For sufficiently large point sets, a proof of a conjecture by Kleitman and Pinchasi is provided. A lower bound of (2t+14n−k(3k+7))/14 is demonstrated for bichromatic lines passing through at most six points. Lower bounds are also established for equichromatic lines passing through at most four, five, or six points.