Numerical determination of the geodesic curves: the solution of a two-point boundary value problem

Document Type: Original Article


1 Department of Surveying Engineering, Arak University of Technology, Arak, Iran

2 Department of Geomatics Engineering, University of Calgary, Calgary, Alberta, Canada


In this paper, we suggest a simple iterative method to find the geodesic path on a surface parameterized by
orthogonal curvilinear system between two given points based on solving Boundary Value Problem. In this
supposed method, an iterative algorithm is used for finding the sufficient initial values as the destination
point agree with the boundary conditions. Geodesic determination between two given points is formulated
for a general surface, and specially tested for reference ellipsoid which has many applications in
geosciences and geodesy. Accuracy of the method is independent on the distant between two points on the
surface. Moreover, it can be used in aviation and sailings for finding the shortest path between start and
destination points.


Main Subjects

Azariadis, P. N., & Aspragathos, N. A. (2001). Geodesic curvature preservation in surface flattening through constrained global optimization. Computer-Aided Design, 33(8), 581-591.
Baeschlin, C. F. (1948). Lehrbuch der Geodasie. Zurich, Orell Fussli, 1948.
Bermejo-Solera, M., & Otero, J. (2010). Global optimization of the Gauss conformal mappings of an ellipsoid to a sphere. Journal of Geodesy, 84(8), 481-489.
Bessel FW (1825) Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen. Astron Nachr 4(86):241–254. [Translated into English by Karney CFF and Deakin REas The calculation of longitude and latitude from geodesic measurements. Astron Nachr 331(8):852–861 (2010)]
Caselles, V., Kimmel, R., & Sapiro, G. (1997). Geodesic active contours. International journal of computer vision, 22(1), 61-79.
Cohen, L. D., & Kimmel, R. (1997). Global minimum for active contour models: A minimal path approach. International journal of computer vision, 24(1), 57-78.
Deschamps, T., & Cohen, L. D. (2001). Fast extraction of minimal paths in 3D images and applications to virtual endoscopy1. Medical image analysis, 5(4), 281-299.
Do Carmo, M. P. (2016). Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition. Courier Dover Publications.
Ghaffari, A. (1971). On the integrability cases of the equation of motion for a satellite in an axially symmetric gravitational field. Celestial mechanics, 4(1), 49-53.
Grafarend, E. W., & Syffus, R. (1995). The oblique azimuthal projection of geodesic type for the biaxial ellipsoid: Riemann polar and normal coordinates. Journal of Geodesy, 70(1-2), 13-37.
Jekeli, C. (2006). Geometric reference systems in geodesy. Division of Geodesy and Geospatial Science, School of Earth Sciences, Ohio State University, 25.
Heitz, S. (1988). Coordinates in geodesy. Berlin; New York: Springer-Verlag, c1988.
Harris, J. W., & Stöcker, H. (1998). Handbook of mathematics and computational science. Springer Science & Business Media.
Karney, C. F. (2013). Algorithms for geodesics. Journal of Geodesy, 87(1), 43-55.
Kasap, E., Yapici, M., & Akyildiz, F. T. (2005). A numerical study for computation of geodesic curves. Applied Mathematics and Computation, 171(2), 1206-1213.
Kimmel, R. (1997). Intrinsic scale space for images on surfaces: The geodesic curvature flow. Graphical models and image processing, 59(5), 365-372.
Kimmel, R., Malladi, R., & Sochen, N. (2000). Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images. International Journal of Computer Vision, 39(2), 111-129.
Krakiwsky, E. J., & Thomson, D. B. (1974). Geodetic position computations (No. NB-DSE-TR-39). Department of Surveying Engineering, University of New Brunswick.
Lantuejoul, C., & Beucher, S. (1980). GEODESIC DISTANCE AND IMAGE-ANALYSIS. Mikroskopie, 37, 138-142.
Lindeberg, T., & ter Haar Romeny, B. M. (1994). Linear scale-space I: Basic theory. In Geometry-Driven Diffusion in Computer Vision (pp. 1-38). Springer, Dordrecht.
Lipschutz, M. M. (1969). Schaum's outline of theory and problems of differential geometry.
Patrikalakis N.M., & Ko K.H., Computational Geometry, Lecture Notes, MIT, 2003.
Rainsford, H. F. (1955). Long geodesics on the ellipsoid. Bulletin Géodésique (1946-1975), 37(1), 12-22.
Sánchez-Reyes, J., & Dorado, R. (2008). Constrained design of polynomial surfaces from geodesic curves. Computer-Aided Design, 40(1), 49-55.
Sjöberg, L. E. (2006). New solutions to the direct and indirect geodetic problems on the ellipsoid. Zeitschrift für Geodasie, Geoinformation und Landmanagement, 131, 1-5.
Shampine, L. F., & Reichelt, M. W. (1997). The matlab ode suite. SIAM journal on scientific computing, 18(1), 1-22.
Thomas, C. M., & Featherstone, W. E. (2005). Validation of Vincenty’s formulas for the geodesic using a new fourth-order extension of Kivioja’s formula. Journal of Surveying engineering, 131(1), 20-26.
Vanicek, P., & Krakiwsky, E. J. (2015). Geodesy: the concepts. Elsevier.
Zhang, L., Zhou, C., Zhang, P., Song, Z., Kong, Y. P., & Han, X. (2010, June). Optimal energy gait planning for humanoid robot using geodesics. In Robotics Automation and Mechatronics (RAM), 2010 IEEE Conference on (pp. 237-242). IEEE.
Zhang, L., & Zhou, C. (2007, May). Robot optimal trajectory planning based on geodesics. In Control and Automation, 2007. ICCA 2007. IEEE International Conference on (pp. 2433-2436). IEEE.