[alogo] 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.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]


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

[0_0]


In the ellipse-equation above a = |R+r|/2, c = |OA|, as before, and b = a - c.
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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