Given the line $x^\alpha=(x,1/2)$ and the parameter $c^\alpha = (0,-1)$, I know that the transformation given by:
$\tilde{x}_1 = \frac{x}{-x^2 + 1/4}$ $\tilde{x}_2 = \frac{x^2+\frac{1}{4}}{\frac{1}{4}-x^2}$
Maps the line to the hyperbola that has an equation of the form:
$y^2 - x^2 = 1$
How do I modify this transformation in order to get the hyperbola corresponding to:
$x^2 - y^2 = 1$ How is in general the method for finding such a transformation for a generic line?
If $f$ is a conformal transformation mapping the stated line to one branch of $y^2 - x^2 = 1$ and if $R$ is a one-quarter turn about the origin (e.g., $R(x, y) = (-y, x)$), then the conformal map $R \circ f$ has the desired effect.
If you're working with a holomorphic map $f$, the holomorphic map $g = if$ has this effect.