Simulate the rendering of a 3D sphere

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Is it possible to efficiently simulate (aka compute) what a texture (simple 2D) image rendered on a sphere and seen from a certain, constant position directly facing the sphere, will look like?

Am looking for an efficient method to achieve a 3D effect based on an image without launching a whole 3D scene and render 3D polygons etc. (mainly because the device does not have hardware acceleration for that).

The effect does not have to be strong (in the end only a part of the sphere would be visible) and not perfect either (target resolution of 128x128. It may (would most likely require) rendering 2D polygons.

Could someone direct me towards the right algorithms? Huge thanks in advance!