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 ElementsEngineering-Scientific Workspace. The latter can plot computed geodesics on 2D 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


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


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 ElementsEngineering-Scientific Workspace.

Related Software

Geodesic computation can be applied to optimization problems.