How can I visually display a custom map projection defined using a math function

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I've been tinkering around with making my own map projections, and I want to test some of the things I'm looking at. I made a desmos graph that distorts a grid according to some function, but I want to be able to actually distort an image of the earth.

I have a function T(x, y), that would take each pixel of an image and move it to some other point defined by the function. I want to be able to do this to an image somehow.

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