Geometric Separators
This program is based off the paper Geometric Separators and the Parabolic Lift by Don Sheehy. It implements a new algorithm to calculate a geometric separator for a set of 2D input points.
About:
This Processing program will allow you to input a number of 2D points and determine a geometric separator for the input set. It will graphically show the centerpoint and the spherical separator projected down to the 2D plane.
How it works:
The points input by the user will be lifted to a paraboloid in 3D. Then, we approximate a centerpoint from sets of 5 points by using Iterated Radon Points. To find a Radon point from a set of 5 points, there are two cases we must test for: 1) if a ray intersects a triangle and 2) if a point is inside a tetrahedron. The intersection point or contained point, respectively, is then the Radon point. These Radon points are reduced down to a single point - our estimated centerpoint. A sphere is then calculated and projected down to the original 2D plane along with the centerpoint.
Usage:
There are three buttons on the application window: Calculate, Reset, and Randomize. You can click anywhere on the canvas to add input points. Then, use the Calculate button, which will show the centerpoint in red (if it can) and the separator projected down to the 2D plane as a gray ellipse. If you don't have specific points, you can use the Randomize button to add 25 random points to the canvas. To remove added points, just click Reset.
Note: The centerpoint is estimated using Radon Points - according to Radon's Theorem, we can only find a partition for sets of 5 points, so we must only have powers of 5 for our input set.
References:
http://donsheehy.net/research/sheehy13geometric.pdf
http://dl.acm.org/citation.cfm?id=161004