8 research outputs found
Finding an ordinary conic and an ordinary hyperplane
Given a finite set of non-collinear points in the plane, there exists a line
that passes through exactly two points. Such a line is called an ordinary line.
An efficient algorithm for computing such a line was proposed by Mukhopadhyay
et al. In this note we extend this result in two directions. We first show how
to use this algorithm to compute an ordinary conic, that is, a conic passing
through exactly five points, assuming that all the points do not lie on the
same conic. Both our proofs of existence and the consequent algorithms are
simpler than previous ones. We next show how to compute an ordinary hyperplane
in three and higher dimensions.Comment: 7 pages, 2 figure
On the number of ordinary conics
We prove a lower bound on the number of ordinary conics determined by a
finite point set in . An ordinary conic for a subset of
is a conic that is determined by five points of , and
contains no other points of . Wiseman and Wilson proved the
Sylvester-Gallai-type statement that if a finite point set is not contained in
a conic, then it determines at least one ordinary conic. We give a simpler
proof of their result and then combine it with a result of Green and Tao to
prove our main result: If is not contained in a conic and has at most
points on a line, then determines ordinary conics.
We also give a construction, based on the group structure of elliptic curves,
that shows that the exponent in our bound is best possible
On the Geometries of Conic Section Representation of Noisy Object Boundaries
This paper studies some geometrical properties of conic sections and the utilization of these properties for the generation of conic section representations of object boundaries in digital images. Several geometrical features of the conic sections, such as the chord, the characteristic point, the guiding triangles, and their appearances under the tessellation and noise corruption of the digital images are discussed. The study leads to a noniterative algorithm that takes advantage of these features in the process of formulating the conic section parameters and generating the approximations of object boundaries from the given sequences of edge pixels in the images. The results can be optimized with respect to certain different criteria of the fittings
On sets defining few ordinary circles
An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least 1/4n2 − O(n) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most 1/24n 3−O(n2) circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most Kn2 ordinary circles, , then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves