Geodesic Surveyor: Measure Distances on Polyhedral Surfaces

This page lets you compute the length of the shortest paths on a polyhedral surface which connects two interactively given points on the surface. The polygonal faces must be triangles.

Select the desired type of geodesic (for description see below or the "Info"-tab) and pick a point on the surface. The values of parameters length and angle always show the length and starting angle of the actual curve you see. If you compute straightest geodesics you can only change the starting point and both parameters length and angle per hand. If you compute shortest geodesics you can only pick either the starting point or the endpoint, but the parameters length and angle will change themselves to the actual length and angle of the shortest geodesic you computed. The parameter angle is the angle regarding the starting triangle, so don't wonder why it jumps whenever you change the triangle.

The concept behind all that happens is that of geodesics from differential geometry: Geodesic curves are (local) shortest curves on a surface and their curvature, measured regarding the surface, vanishes. Now when we have piecewise linear surfaces (so they aren't differentiable)  matters change a bit and lead to two different kinds of geodesics: Geodesics that are local shortest curves (shortest geodesics) and geodesics that have minimal curvature measured on the surface (straightest geodesics). Often shortest geodesics are also straightest geodesics and vice versa, but sometimes they aren't.

• Straightest geodesics: Straightest geodesics are defined in the
  following way: 
  1. At points in the interior of triangles the curve is a straight line.
  2. At points on edges the curve is a straight line if you unfold the surface to a plane.
  3. At points on vertices the angles left and right from the curve (measured on the surface) coincide.

  Straightest geodesics that cross a spherical vertex aren't local shortest curves as you always find a shorter way around the vertex. 
  You can always construct a unique straightest geodesic with given initial point and direction.
  You cannot always connect two arbitrary points on the surface with a straightest geodesic (e.g. if the endpoint lies directly behind a hyperbolic vertex).

• Shortest geodesics: Shortest geodesics are local shortest curves. 
  You can always find a shortest geodesic that connects two arbitrary points.
  Shortest geodesics never cross spherical vertices, so you cannot always extend a shortest geodesic with given initial point and initial direction as shortest even in a small neighbourhood of the initial point.
  Shortest geodesics are not always straightest geodesics (e.g. if they hit hyperbolic vertices) but may have non-zero geodesic curvature.

Detail information about discrete geodesics is given in the paper:

Straightest Geodesics on Polyhedral Surfaces
Konrad Polthier, Markus Schmies
in: Mathematical Visualization, Eds: H.C. Hege, K. Polthier
Springer Verlag, 1998, ISBN 3-540-63991-8, Pages 391.