[lzlpplvi] shortest path on a Euclidean manifold with wormholes

in Euclidean space, the shortest path between two points is a line segment.  then, modify the space:

designate two special points for which travel between them is free, costs no distance, i.e., a wormhole connects the points.  then, the shortest path between two points might no longer be the line between them; it might be to travel to and from the wormhole mouths.

(which pairs of points are affected?  this is a problem in 2*D dimensional space.)

instead of just two special points, add a line of special points with free travel between any two points on the line.  shortest paths will often involve traveling orthogonally to the line, traveling along the line for free, then departing orthogonally from the line to the destination.  maybe this is a river in which transportation cost is negligible.  or, minimize walking where there are moving sidewalks.

the wormhole mouths and river line are examples of sets of points.  a zone of free travel can be generalized to any set of points.  one could have multiple zones: free travel within each zone, but, unless zones are adjacent, you have to exit a zone and travel through the overworld to connect to other zones.

how difficult is finding the shortest path?

once we have a notion of shortest path (geodesic), we can define minimum spanning tree, traveling salesman, Euclidean Steiner tree, Voronoi diagram, Delauney triangulation.