I have multiple faces in 3D space creating cells. All these faces lie within a predefined cube (e.g. of size 100x100x100). Every face is convex and defined by a set of corner points and a normal vector. Every cell is convex. The cells are result of 3d voronoi tessellation, and I know the initial seed points of the cells.

Now for every integer coordinate I want the smallest distance to any face.

My current solution uses this answer https://math.stackexchange.com/questions/544946/determine-if-projection-of-3d-point-onto-plane-is-within-a-triangle/544947 and calculates for every point for every face for every possible triple of this faces points the projection of the point to the triangle created by the triple, checks if the projection is inside the triangle. If this is the case I return the distance between projection and original point. If not I calculate the distance from the point to every possible line segment defined by two points of a face. Then I choose the smallest distance. I repeat this for every point.

This is quite slow and clumsy. I would much rather calculate all points that lie on (or almost lie on) a face and then with these calculate the smallest distance to all neighbour points and repeat this. I have found this Get all points within a Triangle but am not sure how to apply it to 3D space.

Are there any techniques or algorithms to do this efficiently?

CodePudding user response：

Since we're working with a Voronoi tessellation, we can simplify the current algorithm. Given a grid point p, it belongs to the cell of some site q. Take the minimum over each neighboring site r of the distance from p to the plane that is the perpendicular bisector of qr. We don't need to worry whether the closest point s on the plane belongs to the face between q and r; if not, the segment ps intersects some other face of the cell, which is necessarily closer.

Actually it doesn't even matter if we loop r over some sites that are not neighbors. So if you don't have access to a point location subroutine, or it's slow, we can use a fast nearest neighbors algorithm. Given the grid point p, we know that q is the closest site. Find the second closest site r and compute the distance d(p, bisector(qr)) as above. Now we can prune the sites that are too far away from q (for every other site s, we have d(p, bisector(qs)) ≥ d(q, s)/2 − d(p, q), so we can prune s unless d(q, s) ≤ 2 (d(p, bisector(qr)) d(p, q))) and keep going until we have either considered or pruned every other site. To do pruning in the best possible way requires access to the guts of the nearest neighbor algorithm; I know that it slots right into the best-first depth-first search of a kd-tree or a cover tree.