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 Disagreement by Mindrod)

No, the center of your circle radius a does not map to the center of it's image. In fact, no such circle exists (aside from a circle radius 0 centered somewhere on the circle of inversion. With this circle sqrt(b^2 - a^2) = b in which case you'd be right, be we ought to solve more generally.

Also this inversion is a one to one mapping so no two points map to the same point.

Posted by Eric on 2006-02-23 16:13:58