Question

I’m working on the implementation of an algorithm used to determine the safest point for a drone to land, using this paper.

To do so I’m tring to find two parallel planes enclosing a set of 9 points while minimizing the distance r between those two planes.

r will then represent the roughness of the terrain.

I would like a general strategy to solve the problem or a link to a paper describing a solution.

Answer 1

The goal is to find planes normal. Then building the planes is easy.

And there is finite number of candidates for plane normal: cross-products of edge vectors of convex hull (this includes but is not limited to face normals). For this number of points you can just count them all.

Why?

• Every plane touches some non-zero number of points (otherwise it can be moved closer).
• If we can rotate planes even slightly without losing connection with these points, distance will decrease.
• So the optimum planes can not rotate.
• If a plane touches two points, it can rotate only around this edge.
• A plane cannot rotate if it touches two non-parallel edges.
• Then its normal is cross-product of those edge vectors.