Furthermore, the new composite path and the initial path form a triangular region, called geodesic triangle, which satisfies the local Gauss-Bonnet Theorem.

Associated with geodesic replanning procedure, local Gauss-Bonnet Theorem of the geodesic triangle formed by the initial geodesic and the replanned geodesics will also be presented for aiding traversability analysis of the terrain that the replanned path traverses in Section 3.

It is noted that a geodesic triangle [DELTA] [subset] M is formed by the sides composed of the initial geodesic and two replanned geodesics.

Real-time limitation of the replanning algorithm is addressed based on looking at the geometric operations in a coordinate chart such as algorithms used for geodesic computation, intersection checking and the computation of intersection point P of geodesic with circle (see (14)), determination of the tangent direction, critical points determination either via tangent lines to make a directional sweep of the terrain or via search in coordinate parameter space nearby P, projection of the point on tangent plane onto the manifold, collision checking of the new composite geodesic path with the obstacle, and computation of the angle sum of geodesic triangle for applying Gauss-Bonnet Theorem.

A geodesic metric space is called hyperbolic (in the Gromov sense) if there exists an upper bound of the distance of every point in a side of any geodesic triangle to the union of the two other sides (see Definition 2.

We would like to point out that deciding whether or not a space is hyperbolic is usually extraordinarily difficult: Notice that, first of all, we have to consider an arbitrary geodesic triangle T, and calculate the minimum distance from an arbitrary point P of T to the union of the other two sides of the triangle to which P does not belong to.

We consider now a geodesic triangle T in X, a point p [member of] T and 0 < [member of] < 1/2.