# “Shape Aware” Trees

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.

• Inside disks rather than circles... – YCor Mar 25 '18 at 14:59
• @YCor depends on the interpretation of "inside"; if it is interpreted set theoretic in the sense of being an element of the set, then disks is correct. If interpreted geometrically, where the circle is the boundary of a finite region, then one might be tempted to say "inside the circle"; I guess it lies in the eye of the beholder, what seems more appropriate. – Manfred Weis Mar 25 '18 at 18:24
• Actually what's useful in a universal mathematical language is that it does not depend on what's in the eye of the beholder. – YCor Mar 25 '18 at 19:48
• There is no mistaking what was meant. One might substitute "interior to", as when mathematicians refer to an interior region and an exterior region in discussing the Jordan curve theorem. – Todd Trimble Mar 26 '18 at 10:33