convex hull algorithm for 3d surface z = f(x, y)

Question

I have a 3D surface given as a set of triples (x_i, y_i, z_i), where x_i and y_i are roughly on a grid, and each (x_i, y_i) has a single associated z_i value. The typical grid is 20x20

I need to find which points belong to the convex hull of the surface, within a given tolerance. I'm looking for an efficient algorithm to perform the computation (my customer has provided an O(n³) version, which takes ~10s on a 400 point dataset...)

Answer 1

There's quite a lot out there, didn't you search?

Here are a couple with O(n log h) runtime, where n is number of input points and h is number of vertices of the result:

http://en.wikipedia.org/wiki/Chan%27s_algorithm

http://en.wikipedia.org/wiki/Kirkpatrick-Seidel_algorithm

Here is a demonstration of four methods, with links to the algorithms:

http://www.cse.unsw.edu.au/~lambert/java/3d/hull.html

Answer 2

The O(n^3) version is probably Jarvis algorithm for 3d Hull. Look at this algorithm, I think is well described: