A map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is said to be an isometric folding of [[??].sup.4] model into itself if and only if for any piecewise fuzzy

geodesic path [gamma] : J [right arrow] [[??].sup.4] the induced path [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a piecewise fuzzy

geodesic and of the same length as [gamma], where J = [0,1].

The Jacobi field for the Riemannian mean is equal to the sum of tangent vectors to all

geodesics (from mean to each point).

* Draw the arc ADB which is the

geodesic between A and B on the surface of the sphere.

In this paper, by taking both the local features and their connections into account, we propose a novel local shape feature representation method,

geodesic connected graph, for 3D CAD model analysis.

Recall that in every free homotopy class there exists a single simple closed

geodesic, assuming that punctures are simple closed

geodesics with length equal to zero.

As a result, the scalar equation of isotropic

geodesics (the equation of energy), in the "large scale approximation" in the globally non-holonomic space, gives the solution for the photon's energy (frequency) and the redshift z = [[omega].sub.0]-[omega]/[omega] " as depending on the distance r = ct travelled from the observer

Finally, Section 6 discusses very recent results about the structure of

geodesics in the Brownian map.

Since our approach builds on the fact that a wave takes minimum energy path, when the assumption of isotropy in the media is imposed, the shortest energy path reduces to shortest distance path, which is provided by the evaluating the

geodesics. Mathematically, the energy along a path is seen as weighing function defined along the path.

The concept can be extended to apply to open paths in Ray space where |[phi]([tau])> [??] |[phi](0)> by closing the curve by a

geodesic, i.

Sethian and Ron Kimmel, now at the Technion Israel Institute of Technology in Haifa, Israel, described in the July 21, 1998 PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES how to use this method to compute the shortest paths, called

geodesics, from point to point across hilly terrain and other computer-generated landscapes.

So three years ago, Tucker formed Electrical

Geodesics Inc., to develop the

Geodesic Sensor Net.