How to avoid distortion of geodetic shapes mapped on a sphere?

Hello

I can't find how to avoid distortion of shapes rendered in the geodetic projection and mapped on a sphere.
Does anybody know this problem? I would be very grateful for help.

There are some solutions to the problems of this kind in the internet.
Also the attachment contains examples of the particular one
but I have no idea how this trick can be done in GM or another GIS/GDAL tool.

As I suppose the shapes should be rendered somehow as already distorted (streched)
but I don't know how to separate projection of the shapes from projection
of the bounding box of these shapes (let's say, stars)
and, next, how to merge outputs of procedures.

However, with help of professional cartographic tools it should be simpler and easier than I think.
I hope that my ignorance is the major barrier to achieve the target.

Pawel


Three samples of the same area. Left and right ones are originally distorted.
The middle one should also preserve distortion of the geodetic projection to be useful in the project.

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