Given a planar configuration of points in general position. It is known that the Delaunay triangulation is the 'fattest' triangulation possible. It is also easily seen that given 7 points with 6 of them close to the vertices of a regular hexagon and the 7th near its center, there is one triangulation where all 6 resulting triangles are acute and another giving all 6 obtuse triangles.
- Given n planar points, how does one algorithmically achieve a triangulation of n planar points such that (a) the number of acute triangles is maximized (b) the number of obtuse triangles is maximized? What are the best/worst point configurations for these triangulations?
Observation: if the input points lie (say) very close to a circular arc of small angle measure, all triangles in any triangulation are necessarily obtuse. There don't appear to be non-trivial point systems with all triangles in any triangulation being acute.
- Does the Delaunay triangulation also minimize the (a) sum of the lengths of all edges and (b) sum of the diameters of the triangles ? If not, how does one achieve such triangulation(s)?
Note: In all the above, we do not add extra vertices.