How to calculate the medial axis?

Question

Does anyone know how to calculate the medial axis for two given curves?

Medial axis: http://en.wikipedia.org/wiki/Medial_axis

Here is the shape I need to calculate it for:

I drew in the medial axis myself, the dark black line, but I need to be able to calculate it dynamically.

Here is the applet and code of what I have done so far: http://www.prism.gatech.edu/~jstrauss6/3451/sample/

The known variables are: -pt A, B, C, D -radii of red, green, and black circles -pt Q and R (just outside the picture), the black circles.

Answer 1

Let C1 and C2 be centers of circles with radii r1 and r2 . The medial axis (minus the two center points) of the figure made of the two circles is the set of points M satisfying

|M - C1| - r1 = |M - C2| - r2

which implies

|M - C1| - |M - C2| = r1 - r2
|M - C1|^2 + |M - C2|^2 - (r1 - r2)^2 = 2 * |M - C1||M - C2|
(|M - C1|^2 + |M - C2|^2 - (r1 - r2)^2)^2 = 4 * |M - C1|^2 |M - C2|^2  (**)

so the medial axis is a fourth degree algebraic curve.

Let us say that C1 and C2 are on the y axis, and suppose that the point (0,0) lies on the medial axis (so C1 = (0, -r1 - x) and C2 = (0, r2 + x) for some x you can compute from your data). This is something you can always transform into.

Now, you want the curve y = f(x) which parametrizes the median axis. For this, pick the x of your choice, and solve equation (**) in y with Newton's method, with initial guess y = 0 . This is a polynomial you can compute exactly, as well as its derivative (in y ).

Answer 2

The medial axis is in this case a hyberbola.

For more information see this article , particularly the following excerpt:

The center of any circles externally tangent to two given circles lies on a hyperbola, whose foci are the centers of the given circles and where the vertex distance 2a equals the difference in radii of the two circles.

So the problem reduces to drawing a hyperbola, given its foci and vertex distance.

Answer 3

If you embed the circles on a rectangular grid (think image), then you can use the distance transform of this image to compute your medial axis. See this link . Several O(nlogn) algorithms exist for computing the distance map on an image grid.