The point cannot be mapped into a circle. Therefore, the point must be mapped into the center of the concentric circles. The center of inversion cannot lie on the circle or the line since either would be mapped into a line. The center of inversion must lie on the line determined by the point and the center of the circle and this line must be perpendicular to the given line. Therefore, let k be the inversion radius and
Line: x = 0
Point: (c,0)
Circle: (x-b)2 + y2 = a2
Center of inversion: (h,0)
For the circle and the line to be mapped into concentric circles,
k2 k2
--------- + ---------
k2 (b-a)-h (b+a)-h
-------- = -----------------------
2(0-h) 2
or
h = +- sqrt(b2 - a2)
For the point to be mapped into the center of the map of the line,
k2 k2
-------- = -----
2(0-h) c-h
or
h = -c
Therefore, the point must be the reflection about the
line of the center of inversion that maps the circle
and line into concentric circles. With the distance
between the point and line
|c| = sqrt(b2 - a2)
See
www.geocities.com/bractals/l-inv.jpg and
www.geocities.com/bractals/r-inv.jpg
for the two solutions of mapping a circle C, a line L, and a point P into concentric circles C' and L' and their center P' where the circle of inversion I is tangent to the line L. Note circle PI in determining the center of circle I and point P.
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