# Offset approximation of polygons on an ellipsoid

## 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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## Acknowledgements

The authors are very grateful to the anonymous reviewers for their valuable comments and constructive suggestions.

## Author information

Authors

### Corresponding author

Correspondence to Young Joon Ahn.

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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### Keywords

• Geodesic path on ellipsoid
• Offset approximation
• Approximation error
• Implicit algebraic surface
• Oblate and Prolate ellipsoid

• 65D10