How to find the point most distant from a given set and its bounding box

Question

I have a bounding box, and a number of points inside of it. I\'d like to add another point whose location is farthest away from any previously-added points, as well as far away

Answer 1

The name of the problem you're solving is the largest empty sphere problem.

It can easily be solved in O(n^4) time in the plane. Just consider all O(n^3) triples of points and compute their circumcenter. One of these points is your desired point. (Well, in your case, you also have to consider "a side" as one of your three points, so you not only find circumcenters but slightly more general points, like ones equidistant from two points and a side.)

As the Wikipedia link above indicates, the problem can also be solved in O(n log n) time by computing a Voronoi diagram. More specifically, then your desired point is the circumcenter of one of the triangles in the Delaunay triangulation of your points (which is the dual of the Voronoi diagram), of which there are only O(n). (Again, to adapt exactly to your problem, you'll have to consider the effects of the sides of the box.)