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 Inversion Distance (Posted on 2006-02-21)
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.

 See The Solution Submitted by Bractals Rating: 2.8000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re(3): Note to Bractals | Comment 25 of 32 |
(In reply to re(2): Note to Bractals by Mindrod)

` `
`Let O be the center of inversion and P the centerof the circle to be inverted. Let the line OPintersect the circle at points A and B (AB a diameterof the circle). Clearly, f(A)f(B) will be a diameterof the inverted circle. WOLOG let O be the origin (0,0),A (a,0), B (b,0), and P ((a+b)/2,0). So`
`    f(A) is (k^2/a,0)    f(B) is (k^2/b,0)    f(P) is (k^2/[(a+b)/2],0)`
`The center of the inverted circle is`
`      k^2     k^2     ----- + -----       a       b    ---------------           2`
`Therefore, for the center of the the inverted circleto be the same as the image of the center of the givencircle we must have`
`      k^2     k^2     ----- + -----       a       b         k^2    --------------- = ---------           2            a + b                       -------                          2`
`               or`
`    (a - b)^2 = 0`
`Hence, the diameter of the given circle is zero.`
`See www.geocities.com/bractals/c-inv.jpg`
` `

Edited on February 25, 2006, 12:58 pm
 Posted by Bractals on 2006-02-25 12:08:38

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