Is it possible to construct a projection matrix which will create a perspective with four (or more) vanishing points?
This question have an OpenGL tag, but general insights are welcomed as well.
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To the simple question of "can you use a projection matrix" to do this, the answer is no. The reason is that a matrix multiplication can only result in an linear or affine transform that maps lines to lines, whereas you're looking to map a line to a curve.
To the question of "how do I implement a fisheye projection", see: