From ed22d97055915466c48f029a78fb1b1a2cce8015 Mon Sep 17 00:00:00 2001 From: Andrew Lawson Date: Wed, 10 Dec 2014 20:56:34 -0500 Subject: [PATCH] Improved Read me. --- README.md | 12 ++++++++++-- 1 file changed, 10 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index 6e1c066..2e4f03d 100644 --- a/README.md +++ b/README.md @@ -9,13 +9,18 @@ This program is based off the paper [Geometric Separators and the Parabolic Lift About: ------ -This Processing program will allow you to input a number of 2D points and will calculate the geometric separator for the input set. It will graphically show the centerpoint and the spherical separator projected down to the 2D plane. +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](http://dl.acm.org/citation.cfm?id=161004). 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 (if it can) and the separator projected down to the 2D plane. 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**. +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. @@ -23,6 +28,9 @@ References: ---------- http://donsheehy.net/research/sheehy13geometric.pdf + http://dl.acm.org/citation.cfm?id=161004 + http://steve.hollasch.net/cgindex/geometry/ptintet.html + http://geomalgorithms.com/a06-_intersect-2.html \ No newline at end of file