Computation of shortest paths on free-form surfaces is an important problem in ship design, robot motion planning, computation of medial axis transforms of trimmed surface patches, terrain navigation and NC machining. The objective of this paper is to provide an efficient and reliable method for computing the shortest path between two points on a free-form parametric surface and the shortest path between a point and a curve on a free-form parametric surface. These problems can be reduced to solving a two point boundary value problem. Our approach for solving the two point boundary value problem is based on a relaxation method relying on finite difference discretization. Examples illustrate our method.

1.
Beck
J. M.
,
Farouki
R. T.
, and
Hinds
J. K.
, “
Surface Analysis Methods
,”
IEEE Computer Graphics and Applications
, Vol.
6
, No.
12
, pp.
18
36
, December
1986
.
2.
Bliss
G. A.
, “
The Geodesic Lines on the Anchor Ring
,”
Annals of Mathematics
, Vol.
4
, pp.
1
21
, October
1902
.
3.
Dahlquist, G., and Bjo¨rck, A., Numerical Methods, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1974.
4.
do Carmo, P. M., Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.
5.
Ferziger, J. H., Numerical Methods for Engineering Applications, Wiley, 1981.
6.
Keller, H. B., Numerical Methods for Two-Point Boundary Value Problems, Blaisdell, 1968.
7.
Kimmel
R.
,
Amir
A.
, and
Bruckstein
A. M.
, “
Finding Shortest Paths on Surfaces Using Level Sets Propagation
,”
IEEE Transactions on Pattern Analysis and Machine Intelligence
, Vol.
17
, No.
6
, pp.
635
640
, June
1995
.
8.
Kreyszig, E., Differential Geometry, University of Toronto Press, Toronto, 1959.
9.
Mitchell
J. S. B.
, “
An Algorithmic Approach to Some Problems in Terrain Navigation
,”
Artificial Intelligence
, Vol.
37
, pp.
171
201
,
1988
.
10.
Munchmeyer, F. C., and Haw, R., “Applications of Differential Geometry to Ship Design,” in: D. F. Rogers, B. C. Nehring, and C. Kuo, editors. Proceedings of Computer Applications in the Automation of Shipyard Operation and Ship Design IV, volume 9, pages 183–196, Annapolis, Maryland, USA, June 1982.
11.
Patrikalakis
N. M.
, and
Bardis
L.
, “
Offsets of Curves on Rational B-Spline Surfaces
,”
Engineering with Computers
, Vol.
5
, pp.
39
46
,
1989
.
12.
Press, W. H., et al., Numerical Recipes in C, Cambridge University Press, 1988.
13.
Sneyd
J.
, and
Peskin
C. S.
, “
Computation of Geodesic Trajectories on Tubular Surfaces
,”
SAIM Journal of Scientific Statistical Computing
, Vol.
11
, No.
2
, pp.
230
241
, March
1990
.
14.
Struik
D. J.
, “
Outline of a History of Differential Geometry
,”
Isis
, Vol.
19
, pp.
92
120
,
1933
.
15.
Struik, D. J., Lectures on Classical Differential Geometry, Addison-Wesley, Cambridge, Mass., 1950.
16.
Wolter, F.-E., Cut Loci in Bordered and Unbordered Riemannian Manifolds, PhD thesis, Technical University of Berlin, Department of Mathematics, December 1985.
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