Introduction

We consider the following 2D problem: given a test point $A$ on a plane and an ellipse $\mathcal{E}$, find the point of the ellipse which is the closest to the test point.

This problem occurs in several contexts. The first one is to geopositioning. Planet bodies are often roughly modeled as biaxial ellipsoids, i.e. ellipsoids having a rotational symetry axis (the polar axis). For such shapes, the meridian planes are ellipses. Given a point position in 3D cartesian coordinates, we want to compute the longitude, geodesic latitude and altitude. The last two data are obtained by solving this kind of problem. Another context was encountered while computing approximations of graphical representations of 3D circles on a 2D display. In order to build an error model of a Bézier-based approximation of the ellipse, the distance of thousands of approximated curves had to be computed. This later problem is described in another technical note which is available on the same place as this one.

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