ConicsFromCir

Conics from circles

Given two intersecting circles, the geometric locus of points C such that the distances |CD|, |CE| from the two circles (A,r) and (B,R) are equal is a hyperbola together with an ellipse, with foci at the centers of the circles and orthogonal to each other.

In the hyperbola-equation above a = |FG|/2, where FG realizes the difference of the circle radii: |R-r|, whereas b² = c² - a² = |OB|² - |OI|² and c = |OA| is half the distance of the two centers.

In the ellipse-equation above a = |R+r|/2, c = |OA|, as before, and b² = a² - c².
The two conics are orthogonal at their intersection point. In the case of two circles lying outside/inside each other there is only a hyperbola/ellipse satisfying the locus condition. Look at EllipseFromCir.html for the special case of the ellipse.

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