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 re: Disagreement by Eric)

Eric, you said "...this inversion is a one to one mapping so no two points map to the same point. ", which is my point exactly.

We are given a line, a circle, and a point.  Through the process of inversion mapping, we end up with two concentric circles and a point.  It is easy to see that the given point maps to the resulting point; the given line maps to one of the resulting concentric circles (the one whose circumference passes through the center of the "circle of inversion"); and the given circle (with radius a) maps to the other concentric circle.  Because no two points map to the same point, the center of the given circle must be at exactly the same point as the given point for both to map to the same resulting position at the center of the concentric circles. 

And so, since the distance from the center of the given circle to the given line is exactly the same as the distance from the given point to the given line.  That distance is given as b.

Posted by Mindrod on 2006-02-23 21:06:39