Calculating Distance of a Point from an Ellipse Border

I'm thinking about using oriented ellipses to represent curves (dents/bumps etc.) in my physics engine, and have a few questions about working with them:

1. What methods are there to finding the minimum distance between a point and an ellipse? I need methods of varying cost (in terms of # of calculations) for different parts of my engine.

   • I'm currently aware of two methods to testing if a point is inside/outside an ellipse.

     1. In the first you plug in the point coordinates into the equation (x/a)^2 + (y/b)^2 and seeing if it's >, <, or = to 1 (does the output -1 give the min. distance to the ellipse border?)
     2. In the second you translate the point to the ellipse's coordinates and horizontally/vertically stretch both the ellipse and point in order to turn the ellipse into a circle. (I rarely see this method used... any reason I should be aware of?)

2. How do you test the distance between two ellipses? I figure you could combine the two methods above by transforming both ellipses in a way that makes one a circle, then test the distance from the center of the circle ellipse to the regular ellipse's edge, and finally compare that distance to the radius of the circle ellipse.

Solution 1:

Source: Exercise 2.3.18 (p.54) from Convex functions: constructions, characterizations and counterexamples, J.M. Borwein & J.D. Vanderwerff (2010).

Consider $E:=\{(x,y):x^2/a^2+y^2/b^2=1\}$ in standard form. Show that the best approximation is: $$P_E\,(u,v)=\left(\frac{a^2u}{a^2-t},\frac{b^2v}{b^2-t}\right)$$ where $t$ solves $\frac{a^2u^2}{(a^2-t)^2}+\frac{b^2v^2}{(b^2-t)^2}=1$.