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(3): Note to Bractals by Eric)

Eric, as the problem is worded, the line is not mapped to the circle of the standard clock, as in your example, but to the circle mapped by the inversion of the circle of the standard clock. 

As also stated in the problem, the point to be measured to the line is the original point that maps to the point that shares the center of the circle that was mapped by inversion.  As the center
point of the circle mapped by the line must be the same as the center point of the circle mapped by the standard clock (as these two circles are concentric), and as the point mapped by inversion is given as the same point, then the original point must be the same as the center point of the circle of the standard clock.  And, as the distance of the center point of the standard clock is given as b, then the distance from the point  to the line must be b. 

Posted by Dej Mar on 2006-02-25 05:58:05