A circle (of radius a), a line, and a point are mapped by inversion into two concentric circles and the center of those concentric circles. If the distance from the given circle's center to the line is b, then what is the distance from the point to the line?

Inversion Defined:

Let O be the center of a circle of radius k. An inversion with respect to circle O is a mapping f:R² -> R² such that for all P in R² (not O), P' = f(P) lies on ray OP and
|OP'||OP| = k².

See www.geocities.com/bractals/inv.jpg

for graphical description of inversion.

(In reply to special case? spoiler, perhaps by Mindrod)

The flaw in this logic is thinking that the center of a circle outside maps to the center of a circle inside.

To see this, construct the inverse circle and an interior and exterior circle which map to one another. then draw the rays which are tangent to both circles and pass through O. Now draw the arcs connecting these tangents. You'll see that the centers of these two circles are both on the same side of these arcs whereas the inner part of one circle maps to the outer part of the other meaning the centers cannot map to each other.

Edited on February 22, 2006, 2:05 am

Posted by Eric on 2006-02-22 01:50:11