# How to define the transform?

$y=f(x)$ 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 ?

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.
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
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