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

Torus (elliptical cross section) with Geodesic

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

Salient Assemblage with Geodesic

Close-up

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.

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:

• 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

• Elements™ Engineering-Scientific Workspace.
• EleNTorus™ computes n-sphere, n-ellipsoid, and n-torus.
• EleSalient™ software computes salient assemblages (recursive bumps) on n-dimensional manifolds, including all EleNTorus manifolds.
• EleGrid™ - grid generator with recursive refinement.
• EleRender™ - surface renderer with geodesics and scalar field shading.

Geodesic computation can be applied to optimization problems.