Point distance from ellipse outline

Distance from a Point to an Ellipse in 2D

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Original post by CloudNine Hi, This is in a similar vein to my previous thread, involving GDI+ and ellipses (here). In short, I'd like to know a general formula for a 2 dimensional point's (shortest) distance from a ellipse's outline at any angle (if I've got the x,y, width and height of the ellipse).

The first step is to transform the point into the local space of the ellipse. This reduces the problem to that of finding the distance between a point and an axis-aligned ellipse centered at the origin.

The problem can be solved by finding the points on the ellipse for which the normal at that point is parallel to the vector from the point to the query point. The closest of these can then be used to compute the distance.

This reduces to solving a quartic equation. For details, see this paper. You can find example source code at the same site (geometrictools.com).

[Edit: Beaten. ]

Thanks for the linky - it's helpful! I'd just like to clarify some of the terms used in the example code - such as "extent". Would the extent of my ellipse be "width/2" and "height/2" in my case? It's definitely a harder task than I thought though.

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Original post by CloudNine Would the extent of my ellipse be "width/2" and "height/2" in my case?

Yes. An input to the distance function is an Ellipse2 object. Look at the comments in the header file Wm3Ellipse2.h (on the Mathematics page of that web site).