Problems and Results in Discrete and Computational Geometry
Justin W. Smith · OhioLink ETD Center (Ohio Library and Information Network) · 2012
Let S be a set of n points in 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 + kFor similar conditions and sufficiently large n, (inspired by the work of P. D. T. A. Elliott in [1]) we also show that the number of spheres determined by n points is at least 1 + n-1 3 -t orchard 3 (n-1), and this bound is best possible under its hypothesis.(By t orchard 3 (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.We demonstrate an infinite family of pseudoline arrangements each with no member incident to more than 1 9 (4n -10) points of intersection, where n is the number of pseudolines in the arrangement.We also prove a generalization of the Weak Dirac that holds for more general incidence structures.4.4 The wedge for j = 1, the base case for our induction. . . . . . . .4.5 The arrangement for j = 1, containing 3(6j + 2) + 1 = 25 pseudolines, each of which incident to at most 10 vertices. . . . . . .4.6 The wedge for j = 2. . . . . . . . .