Assume a set of evenly distributed points on a sphere. Each point is contained in a plane normal to the sphere. The planes will intersect each other forming convex polygons centered on the points.
How do I find the corner points of the polygons?
I can generate a set of points that are roughly evenly distributed on a sphere. Using POV-Ray, I can generate a larger sphere, then 'cut-away' the planes that are normal to those points. I am left with a 3D shape made of convex polygons. Such a shape can be seen at https://i.stack.imgur.com/XZCic.jpg
I want to know the corner points of the polygons so I can create a 2D drawing of each polygon; which will then be used to carve plywood "faces", to build a large version of the shape.
Each corner point resides at the intersections of three adjoining polygons; and the polygons are contained in planes that are normal to the sphere. So it seems likely I need to determine the formulas for all the planes, then calculate the intersections, three planes at a time, to find the corners of the polygons.
How do I find the intersections of three planes, given that I know a point in each plane, and that the planes are normal to a sphere, and the points are on the surface the sphere?
The explanation at vi tutor -- http://www.vitutor.com/geometry/space/three_planes.html -- probably explains it, but I can't follow the explanation. Also the Wikipedia article on Plane_(geometry) probably explains some of what I need, but I can't follow its explanations either.
Assuming the intersection of three planes will give me the corner points, this generates the separate question of determining which points are "neighbors" (their polygons share a border); otherwise I will generate a vast number intersections -- only a few of which are of interest.
Note that the planes normal to the sphere at two (non-antipodal) points intersect on a line that projects to the great circle equidistant between those two points. So what you need is a Voronoi diagram on the sphere: the corner points correspond to the intersections of three planes corresponding to Voronoi regions with a common vertex.