EleGeodesic approximates a minimum-length geodesic between given endpoints on a multidimensional manifold (surface).
It returns the shortest path found.
For a complete description, access Netlib and read comments in source code.
EleGeodesic is implemented in Elements Engineering-Scientific Workspace.
The latter can plot computed geodesics on 2D surfaces.
Sphere with Geodesic
Torus (elliptical cross section) with Geodesic
Geodesic on helicoid color-coded by mean-curvature showing sign change.
Salient Assemblage with Geodesic
Close-up from different view point
Close-up from different view point
Geodesic Depends on Metric
Euclidean metric ds=sqrt(dx2+dy2+dz2) on sphere produces a "great circle" geodesic.
Hyperbolic metric ds=|1/z|sqrt(dx2+dy2+dz2) means distance measurement varies inversely with proximity to "equator" plane.
It produces a very different geodesic.
EleGeodesic can compute both initial-value and 2-point boundary-value geodesic problems.
The above are 2-point problems.
The initial-value problem allows multiple self intersections.
And here is the same geodesic plotted in parameter space.
Here is a geodesic trajectory on a torus that never crosses the inner annulus.
Also, it is an approximately closed geodesic.
Here is a geodesic trajectory on an ellipsoid color-coded by Gaussian curvature.
EleGeodesic software and its support has contributed to the following applications.
- Tent manufacturing.
- Cutting and painting path.
- Fiberglass tape windings in pipe manufacturing.
- Equally spaced points on a sphere.
- Optical light path in a curved surface.
- Curved surface (non-developable) textile manufacturing.
- Optimization steepest-path computation.
Geodesic slide show covers many interesting aspects of geodesics.
A version of EleGeodesic called Geodes is in the Netlib repository of numerical software
administered by Oak Ridge National Laboratory, Mathematical Science Section and AT&T Bell Laboratories.
Netlib access to Geodes
Geodes can be extended by:
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- Allowing indefinite metric with null geodesic(s).
- Optimizing grid points dynamically. Using recursion.
- Optimizing performance control values dynamically.
- Finding families of geodesics.
- Add localized salients (bumps) to the manifold.
- Plotting geodesic in both parameter and ambient space.
Some of these capabilities are implemented in Elements Engineering-Scientific Workspace.
Geodesic computation can be applied to optimization problems.