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 Circular map (Posted on 2006-11-15)
Let f be a one-to-one correspondence of the points in a plane. Prove or disprove the following statement:

"If f maps circles to circles, then it maps straight lines to straight lines."

 See The Solution Submitted by JLo No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 re(5): Full Proof | Comment 24 of 29 |
(In reply to re(4): Full Proof by Joel)

`Hi Joel. Thanks for the input.`
`I think my confusion stems from the mappingof the y-axis to the y-axis. `
`Let P and P' be abstract planes and A and Bdistinct points in P.`
`We want to show that `
`     f[Line(A,B)] = Line(f(A),f(B))`
`                  or`
`     Line(A,B) = g[Line(f(A),f(B))`
`Note: I am using g for f^(-1).`
`I will stipulate that `
`     Line(f(A),f(B)) is a subset of f[Line(A,B)].`
`Could you please rewrite your proof so that`
`Points are denoted by A, B, C, ... or                       f(A), f(B), f(C), ... or                      g(A), g(B), g(C), ...`
`A circle is denoted by Circle(X,Y,Z) where  X, Y, and Z are non-collinear points. Hopefully,  from the points I can determine whether the  circle is in P or P'.`
`Note: If Circle(X,Y,Z) is in P, then f[Circle(X,Y,Z)]      = Circle(f(X),f(Y),f(Z)) is a circle in P'.`
`Note: If Circle(X,Y,Z) is in P', then g[Circle(X,Y,Z)]      might or might not equal Circle(g(X),g(Y),g(Z))      in P.`
` `

 Posted by Bractals on 2006-12-18 12:53:22

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