EleGeodesic Software

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.

Surfaces

Sphere with Geodesic

Geodesic on Sphere

Torus (elliptical cross section) with Geodesic

Geodesic on Torus

Geodesic on helicoid color-coded by mean-curvature showing sign change.

Geodesic on Helicoid

Salient Assemblage with Geodesic

Geodesic on Salient

Close-up

Geodesic on Salient

Close-up from different view point

Geodesic on Salient

Close-up from different view point

Geodesic on Salient

Geodesic Depends on Metric

Euclidean metric ds=sqrt(dx2+dy2+dz2) on sphere produces a "great circle" geodesic.

Euclidean Metric

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.

Hyperbolic Metric

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.

Initial-Value Problem

And here is the same geodesic plotted in parameter space.

Parameter Space

Here is a geodesic trajectory on a torus that never crosses the inner annulus. Also, it is an approximately closed geodesic.

Geodesic Trajectory on Torus

Here is a geodesic trajectory on an ellipsoid color-coded by Gaussian curvature.

Geodesic Trajectory on Ellipsoid

Applications

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.

Slide Show

Geodesic slide show covers many interesting aspects of geodesics.

Netlib Access

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.

Related Software

Geodesic computation can be applied to optimization problems.