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Circular map (Posted on 2006-11-15) Difficulty: 5 of 5
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    
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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 mapping
of the y-axis to the y-axis.
Let P and P' be abstract planes and A and B
distinct points in P.
We want to show that 
     f[Line(A,B)] = Line(f(A),f(B))
     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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