Thanks for the suggestion, but I would really like to do this with Painter only. (Unfortunately, my brain does not have an OpenGL shaders -module installed. )
As the sectors in the fan are really less than one degree wide each, it would not hurt to visualize them as a set of narrow triangular wedges, each showing one pixel wide stripe from the original image. So the pixels in the image would be represented as parallel lines with their length increasing while getting further from the center.
Is there a function inside the Painter that can render an Image in a way that each one of the Image corners will land in independently specified coordinates and everything in between simply gets interpolated evenly therein?