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.

Motivation:

This Processing program will allow you to input a number of 2D points and determine a geometric separator for the input set. This lets us tackle geometric divide-and-conquer problems by dividing our input set nicely. According to the process presented in the paper, 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 iteratively reduced down to a single Radon point, which is 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 - these appear in black. If you don't have specific points in mind, you can use the Randomize button to add 25 random points to the canvas. To remove added points, just click Reset. Once finished, click 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. All points that are included in the separator will be highlighted in green.

Note: The centerpoint is estimated using Radon Points (as mentioned earlier) - 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.