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$y=f(x)$ is continuous and defined for all $x$ real numbers.

Point $A(0,f(0))$ is to be moved to on x axis while $f(x)$ is rigid curve and the rigid curve always passes on point $B (x_1,f(x_1))$ as well. How can the transformed new curve be defined ?

Thanks a lot for answers and advice how to start.

EDIT: An Example: if $f(x)=ax$ then after the transformation : $f(x)$ will turn $g(x)=ax_1(x−α)/(x_1−α)$ where $α$ is transformation parameter.

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I'm having trouble interpreting the second sentence, here.... –  Cameron Buie Jun 17 '12 at 14:43
    
Could you give a basic example, say, with a straight line? I can't get the construction either. Is $x_1$ fixed? –  Pedro Tamaroff Jun 17 '12 at 14:44
    
I have added a picture to show what I mean. f(x) is a rigid curve and A is to be moved to A' and curve will passes from B as well –  Mathlover Jun 17 '12 at 15:01
    
I think I got it. It's like a pursuit curve of some sort. –  Pedro Tamaroff Jun 17 '12 at 15:04
    
If you want to apply rigid transformations to a curve such that it passes through $A'$ and $B$, you only have to pick any two points $X$, $Y$, such that $d(X, Y) = d(A', B)$ and the transformation will exist. Is there some constraint I'm missing? –  Karolis Juodelė Jun 17 '12 at 15:17

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