# Plane transformation: intersecting lines to parallel lines

Is there a plane transformation that will convert intersecting lines to parallel and vice versa?

The problem is in converting a trapezoid to a rectangle:

So that means that I have to find a transforamtion for left and right sides of the trapezoid to make them left and right sides of the rectangle.

Consider that lines are $x+y-1=0$, $-x+y-1=0$. And we have to transorm them to $x=-1$ and $x=1$ respectively. I tried Möbius transformation $f(z) = \frac{i (-i+z)}{i+z}$ but it converted parallel lines to circles.

Also I tried to find such functions $\varphi(x,y)$, $\psi(x,y)$, that $x=\varphi(x',y')$, $y=\psi(x',y')$. But subsituting into the formulas I get $\varphi(x',y') = x'+y'$, which does not satisfy the equation for the second line.

I think there is a beautiful transformation that converts lines to lines, not circles, but what is it?