Abstract
We present an approximation method for offset curves of polygons on an oblate ellipsoid using implicit algebraic surfaces. The polygon on the ellipsoid is given by a set of vertices, i.e. points on the ellipsoid. The edges are the shortest geodesic paths connecting two consecutive points. The offset curve of the polygon consists of two parts. The offset curve of an edge is the set of points that are at the same distance from each point on the edge, in the normal direction of the edge. The offset curve of a vertex is the set of points that are at the same geodesic distance from the vertex. Our offset approximation method uses plane section curves for offset curves of edges and prolate ellipsoids for offset curves of vertices. Since our offset approximation curve is constructed from implicit algebraic surfaces, it is easy to check whether a given point on the oblate ellipsoid has intruded into the inside of the offset curve. Moreover our method achieves extremely small approximation errors. We apply our method to numerical examples on the Earth.
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References
Ahn YJ, Cui J, Hoffmann CM (2011) Circle approximations on spheroids. J Navig 64:739–749
Ahn YJ, Hoffmann CM (2018) Sequence of \({G}^n\) LN polynomial curves approximating circular arcs. J Comput Appl Math 341:117–126
Ahn YJ, Hoffmann CM (2019) An approximation of geodesic circle passing through three points on an ellipsoid. J Appl Geodesy 13:329–333
Bartoň M, Hanniel I, Elver G, Kim M-S (2010) Precise Hausdorff distance computation between polygonal meshes. Comput Aided Geom Desi 27:580–591
Farouki RT, Sakkalis T (1990) Pythagorean hodographs. IBM J Res Dev 55:637–647
Farouki RT, Srinathu J (2017) A real-time CNC interpolator algorithm for trimming and filling planar offset curves. Comput Aided Des 86:1–11
Feldman MB, Colson D (1981) The maritime boundaries of the United States. Am J Int Law 75(4):729–763
Gilbertson CP (2011) Earth section paths. Navig: J Inst Navig 59:1–7
Hoffmann CM (1990) Algebraic and numerical techniques for offsets and blends. In: Computation of curves and surfaces. Springer, Dordrecht, pp 499–528
Hoschek J (1985) Offset curves in the plane. Comput Aided Des 17:77–82
Kallay M, Porobic D (2010) Buffer with arcs on a round earth. In: Sixth international conference on geographic information science. Article 134
Karney CF (2013) Algorithms for geodesics. J Geod 87:43–55
Karney CF (2019) GeographicLib, version 1.50.1. https://geographiclib.sourceforge.io/html
Kim Y-J, Oh Y-T, Yoon S-H, Kim M-S, Elber G (2010) Precise haussdorff distance computation for planar freeform curves using biarcs and depth buffer. Vis Comput 26:1007–1016
Lee IK, Kim MS, Elber G (1996) Planar curve offset based on circle approximation. Comput-Aided Des 28:617–630
Pham B (1992) Offset curves and surfaces: a brief survey. Comput Aided Des 24:223–229
Sitepu MJ (1998) Maritime boundary delimitation the Indonesian case. J Coast Dev 1(3):235–243
Symmons CR (1980) The Canadian 200-mile fishery limit and the delimitation of maritime zones around St. Pierre and Misquelon. Ottawa L. Rev. 12:145–165
Wein R (2007) Exact and approximate construction of offset polygons. Comput Aided Des 39:1518–527
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The authors are very grateful to the anonymous reviewers for their valuable comments and constructive suggestions.
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This study was supported by research funds from Chosun University, 2019, and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03032504)
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Ahn, Y.J., Hoffmann, C.M. Offset approximation of polygons on an ellipsoid. Acta Geod Geophys 56, 293–302 (2021). https://doi.org/10.1007/s40328-021-00335-7
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DOI: https://doi.org/10.1007/s40328-021-00335-7
Keywords
- Geodesic path on ellipsoid
- Offset approximation
- Approximation error
- Implicit algebraic surface
- Oblate and Prolate ellipsoid