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How to map a convex quadrilateral onto a square so that lines would transform into lines?

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    Do you want it to send the vertices to the vertices? That won't exist for a general convex quadrilateral. – 2017-01-26
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    I used the wrong words for the search before I posted the question. See the answer – 2017-01-26
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    I see - if you can't edit the title then at least change the body of your question (that you want a *projective*, not a conformal map) – 2017-01-26

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The answer is Patapom's $(u, v) \rightarrow (x, y)$ transform here:
Mapping Irregular Quadrilateral to a Rectangle