A geodesic triangle [DELTA](p, q, r) in a geodesic space (X, d) consists of three points p, q, r in X (the vertices of [DELTA]) and a choice of three geodesic segments [p, q], [q, r], [r, p] (the edge of [DELTA]) joining them.

Suppose that [DELTA] is a geodesic triangle in (X, d) and [bar.[DELTA]] is a comparison triangle for [DELTA].

Gromov-see, e.g., [2], page 159) if it is geodesically connected, and if every geodesic triangle in X is at least as "thin" as its comparison triangle in the Euclidean plane.

A geodesic triangle [DELTA] ([x.sub.1], [x.sub.2], [x.sub.3]) in a geodesic metric space (X, d) consists of three points [x.sub.1], [x.sub.2], [x.sub.3] in X (the vertices of [DELTA]) and a geodesic segment between each pair of vertices (the edges of [DELTA]).

is a CAT(0) space if every geodesic triangle [DELTA] in X satisfies the CAT(0) inequality; namely, the distance between any two points of such a triangle is less than or equal to the distance between the corresponding points of the model triangle in the Euclidean space, that is, a triangle [bar.[DELTA]] with sides of the same length as the sides of [DELTA] and then of the same perimeter.

A geodesic triangle [DELTA]([x.sub.1], [x.sub.2], [x.sub.3]) consists of three vertices [x.sub.1], [x.sub.2], [x.sub.3] [member of] X and three geodesic segments joining each pair of vertices.

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.

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 2.1).

If [x.sub.1],[x.sub.2],[x.sub.3] [member of] X, a geodesic triangle T = {[x.sub.1],[x.sub.2],[x.sub.3]} is the union of three geodesics [[x.sub.1],[x.sub.2]], [[x.sub.2],[x.sub.3]] and [[x.sub.3],[x.sub.1]].

Let [delta] = ([P.sub.0], [P.sub.1], [P.sub.2]) be a

geodesic triangle in M, with vertices [P.sub.0], [P.sub.1], [P.sub.2], corresponding edges [[gamma].sub.0], [[gamma].sub.1], [[gamma].sub.2] and angles [[alpha].sub.0], [[alpha].sub.1] and [[alpha].sub.2].

Let [DELTA] be the

geodesic triangle with vertices x, y, z and let I denote W [intersection] [DELTA].