Geodesic Intersections
Publication: Journal of Surveying Engineering
Volume 150, Issue 3
Abstract
A complete treatment of the intersections of two geodesics on the surface of an ellipsoid of revolution is given. With a suitable metric for the distances between intersections, bounds are placed on their spacing. This leads to fast and reliable algorithms for finding the closest intersection, determining whether and where two geodesic segments intersect, finding the next closest intersection to a given intersection, and listing all nearby intersections. The cases where the two geodesics overlap are also treated.
Practical Applications
The intersection of lines plays a central role when performing geometric operations on geographical objects. Often, this is performed on a map projection; but this has the disadvantage that the projection introduces an inevitable distortion. It is, therefore, preferable to compute the intersections directly on the surface of the Earth (or, more precisely, on some ellipsoidal approximation to the Earth); in this case, the lines in question are best taken to be geodesics, the generalization of straight lines to a curved surface. This paper describes a fast, reliable, and accurate method of computing the intersections of geodesics.
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Data Availability Statement
The source code for the C++ implementation of the intersection routines is available at https://github.com/geographiclib/geographiclib/tree/r2.3. This is documented in Karney (2023).
References
Baselga, S., and J. C. Martínez-Llario. 2018. “Intersection and point-to-line solutions for geodesics on the ellipsoid.” Stud. Geophys. Geod. 62 (3): 353–363. https://doi.org/10.1007/s11200-017-1020-z.
Botnev, V. A., and S. M. Ustinov. 2015. “Metodika opredeleniya rasstoyaniya mezhdu tochkoy i liniyey v geodezii (Method for finding the distance between a point and a line in geodesy).” St. Petersburg State Polytech. Univ. J. 6 (234): 33–44. https://doi.org/10.5862/JCSTCS.234.4.
Botnev, V. A., and S. M. Ustinov. 2019. “Metodika opredeleniya rasstoyaniya ot tochki do otrezka v zadachakh navigatsii (Distance finding method between a point and a segment in navigation).” St. Petersburg State Polytech. Univ. J. 12 (2): 68–79. https://doi.org/10.18721/JCSTCS.12206.
Karney, C. F. F. 2013. “Algorithms for geodesics.” J. Geod. 87 (1): 43–55. https://doi.org/10.1007/s00190-012-0578-z.
Karney, C. F. F. 2023. “GeographicLib, version 2.3.” Accessed January 15, 2024. https://geographiclib.sourceforge.io/C++/2.3.
Karney, C. F. F. 2024. “Geodesics on an arbitrary ellipsoid of revolution.” J. Geod. 98 (1): 4. https://doi.org/10.1007/s00190-023-01813-2.
Martínez-Llario, J. C., S. Baselga, and E. Coll. 2021. “Accurate algorithms for spatial operations on the spheroid in a spatial database management system.” Appl. Sci. 11 (11): 5129. https://doi.org/10.3390/app11115129.
Sjöberg, L. E. 2008. “Geodetic intersection on the ellipsoid.” J. Geod. 82 (9): 565–567. https://doi.org/10.1007/s00190-007-0204-7.
Todhunter, I. 1886. Spherical trigonometry. 5th ed. London: Macmillan.
Information & Authors
Information
Published In
Journal of Surveying Engineering
Volume 150 • Issue 3 • August 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Aug 1, 2023
Accepted: Jan 19, 2024
Published online: Mar 30, 2024
Published in print: Aug 1, 2024
Discussion open until: Aug 30, 2024
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