Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-04-30T17:25:17.884Z Has data issue: false hasContentIssue false
  • English
  • Français

Circle Approximations on Spheroids

Published online by Cambridge University Press:  12 September 2011

Young Joon Ahn ,
Jian Cui  and
Christoph Hoffmann
Young Joon Ahn*
Affiliation:
(Chosun University, South Korea)
Jian Cui
Affiliation:
(Purdue University, USA)
Christoph Hoffmann
Affiliation:
(Purdue University, USA)
*
(Email: ahn@Chosun.ac.kr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We present an approximation method for geodesic circles on a spheroid. Our ap­proximation curve is the intersection of two spheroids whose axes are parallel, and it interpolates four points of the geodesic circle. Our approximation method has two merits. One is that the approximation curve can be obtained algebraically, and the other is that the approximation error is very small. For example, our approximation of a circle of radius 1000 km on the Earth has error 1·13 cm or less. We analyze the error of our approximation using the Hausdorff distance and confirm it by a geodesic distance computation.

Keywords

geodesic circlespheroidgeodesic distancegeodesic path
Type
Research Article
Information
The Journal of Navigation , Volume 64 , Issue 4 , October 2011 , pp. 739 - 749
Copyright
Copyright © The Royal Institute of Navigation 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ahn, Y. J.. (2005) Helix approximation with conic and quadratic Bezier curves. Comp. Aided Geom. Desi., 22, 551–565.CrossRefGoogle Scholar
[2]Ahn, Y. J.. (2010) Approximation of conic sections by curvature continuous quartic Bezier curves. Comp. Math. Appl., 60, 1986–1993.CrossRefGoogle Scholar
[3]Barton, M., Hanniel, I., Elver, G., and Kim, M.-S.. (2010) Precise Hausdorff distance computation between polygonal meshes. Comp. Aided Geom. Desi., 27, 580–591.CrossRefGoogle Scholar
[4]Earle, M. A.. (2000). A vector solution for navigation on a great ellipse. The Journal of Navigation 53, 473–481.CrossRefGoogle Scholar
[5]Earle, M. A.. (2008). Vector solution for azimuth. The Journal of Navigation 61, 537–545.CrossRefGoogle Scholar
[6]Gonzalez, A. R.. (2008). Vector solution for the intersection of two circles of equal altitude. The Journal of Navigation, 61, 355–365.CrossRefGoogle Scholar
[7]Hoffmann, C. M.. (1989). Geometric and Solid Modeling. Morgan Kaufmann, San Mateo, CA, 1989. http://www.cs.purdue.edu/homes/cmh/distribution/books/geo.html.Google Scholar
[8]Kallay, M.. (2007). Defining edges on a round earth. In ACM GIS '07, 2007. Article 63.CrossRefGoogle Scholar
[9]Kallay, M.. (2008). Geometric algorithms on an ellipsoid earth model. In ACM GIS '08, 2008. Article 42.CrossRefGoogle Scholar
[10]Kim, Y.-J., Oh, Y.-T., Yoon, S.-H., Kim, M.-S., and Elber, G.. (2010). Precise Haussdorff distance computation for planar freeform curves using biarcs and depth buffer. The Visual Computer, 26, 1007–1016.CrossRefGoogle Scholar
[11]Pallikaris, A. and Latsas, G.. (2009). New algorithm for great elliptic sailing (GES). The Journal of Navigation 62, 497–507.CrossRefGoogle Scholar
[12]Patrikalakis, N. M. and Maekawa, T.. (2002). Shape Interrogation for Computer Aided Design and Manufacturing. Springer Verlag, New York.CrossRefGoogle Scholar
[13]Rollins, C.. An integral for geodesic length. (2010). Survey Review, 42, 20–26.CrossRefGoogle Scholar
[14]Schechter, M.. (2007). Which way is Jerusalem? Navigating on a spheroid. The College Mathematics Journal, 38, 96–105.CrossRefGoogle Scholar
[15]Sjöberg, L.. (2002). Intersection on the sphere and ellipsoid. J. Geod, 76, 115–120.Google Scholar
[16]Sjöberg, L.. (2007). Precise determination of the Clairaut constant in ellipsoidal geodesy. Survey Review, 39, 81–86.CrossRefGoogle Scholar
[17]Sjöberg, L.. (2008). Geodetic intersection on the ellipsoid. J. Geod, 82, 565–567.CrossRefGoogle Scholar
[18]Tseng, W.-K. and Lee, H.-S.. (2010). Navigation on a great ellipse. J. Mari. Scie. Tech., 18, 369–375.Google Scholar