I have a set of points on the infinite (well, double precision) 2D plane.
Given the convex hull for this set, how can find some points on the inside of the convex hull that are relatively far away from all the points in the input set?
In the image below, the black points are part of the original set and the hatched area represents the space taken up by all the points if we “grow” them with radius R.
The orange points are examples of what I’d like to get. It doesn’t really matter where exactly they are, as long as they are relatively far away from all the black points.
Furthest Point Search http://en.wiki.mcneel.com/content/upload/images/point_far_search.png
Update: Using a delaunay algorithm to find large empty triangles seems to be a great approach for this:
Delaunay Solution http://en.wiki.mcneel.com/content/upload/images/DelaunaySolutionToInternalFurthestPoints.png
This is a good example of a problem that may be solved using KD-Trees… there are some good notes in Numerical Recipes 3rd Addition.
If you are trying to find point locations that are relatively isolated… maybe the center of the largest quad elements would be a good candidate.
The complexity would be O(n log^2 n) for creating the KD-Tree… and creating a sorted list of quad sizes would be O(n Log n). Seems reasonable for even a large number of points (of course, depending on your requirements).