How to calculate the "radius" of an arbitrary point on an ellipse

I'm having a devil of a time trying to get my head around this.
Here is the scenerio:

Tie the ends of a length of string together, forming an infinite loop.
Place the string over a 2x6 vertical stud.

Using a pencil, placed inside the string loop, and pulling the pencil away from the 2x6,
describe a line on the floor.

You will draw an ellipse on the floor, like below.
(The proportions in my drawing are a bit off, but it will still be an ellipse.)

x, y, and p are known.

So here is the question:
how can I calculate the "radius", so to speak,
of vectors a, b, c or any other arbitrary point along the ellipse?

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I'm not sure you can arbitrarily find the radii of an ellipse. Even worse, a free hand ellipse.

All the equations I know depend on r[SUB]1[/SUB] and r[sub]2[/sub] being know.

There may be a work around if you can calculate the area and/or circumference and then "reverse engineer" the equations, but don't know how reliable that will be.

If you find a way, be sure to post it and good luck.

The general equation is:

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a and b are some constant from the focal point.

and the string method:

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In viewing the above from Kelly I was struck by the fact that in the top equation a and b represent 1/2 of the x and y axes distance from the extreme dimensions of an ellipse; and as a result this equation is totally useless for the purpose of identifying the critical elements of the distance between the pins and length of the string required to achieve an ellipse with the desired a and b dimensions. For those interested in how this can be done without any real mathematics take a look at:

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