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

enter image description here

The problem is in converting a trapezoid to a rectangle: enter image description here

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?

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Of course there is, a projective transformation that transforms the point of intersection to the infinity will do the trick. But then you will need to use homogeneous coordinates to take into consideration the line at infinity.

Note that the actual transformation is not unique. You will just have to specify a fundamental set of four points. enter image description here

The choice is more obvious in your quadrilateral case...

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