I want to put / wrap images in 3D objects. To make everything simple and fast, instead of using (and learning) a 3D library, I want to use cartographic images. Image mapping is used this way:

So, you create display images once for each object and use the same mapping for all the images you want to wrap.
My question is, how can I generate such cartographic images (given the 3D model)? Since I do not know the terminology, my searches failed me. Sorry if I use the wrong jargon.
Below you can see a description of the workflow.

I have a 3D model of the object and the input image, I want to generate map images that I can use to create a textured image.
I don’t even know where to start, any pointers are evaluated.
Additional Information
My initial idea was to somehow connect identity mappings (see below) using an external program. I created horizontal and vertical gradient images in Photoshop to see if mapping works using Photoshop images. The result does not look very good. I did not hope, but it was worthy.
entrance

displays (x and y), they just resize the image, they do nothing.


result

As you can see, there are many artifacts. The custom map images that I generated by warping the gradients even look worse.
Here is more information on mappings: http://www.imagemagick.org/Usage/mapping/#distortion_maps
I am using the OpenCV remap () function for matching.
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