Have these kinds of geometric spanning trees already been described?
all three are Minimum Spanning Trees of points that are elements of open disks, the only difference being in how the edge-weights are calculated:
- the top tree is the ordinary MST with euclidean distance between the points resembling the vertices
- the lower left uses the maximal angle, under which an edge is seen from the disks boundary
- the lower right one uses the smaller of the local maxima of the angles under which an edge is seen from the disks boundary
The two lower trees are in some sense "orthogonal" to each other in that the edges of the left one "strive" for orthogonality to the boundary, whereas the edges of the right one "strive" for being parallel to the disks boundary and both trees "try" to accomplish that with "short" edges.
I can imagine, that a potential use could be in the tesselation of regions for use in finite elements methods.