I'm trying to understand the proof of the theorem (given in the link) that states "Stereographic projection maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane."
In the proof it states "In order to obtain an equation for the projection points (x, y) ∈ C of the circle c under stereographic projection, we substitute (1) into Equation (2), which yields"
Why does plugging in the pre image of points from the image plane into an arbitrary plane give me an equation of the points under stereographic projection? What is the significance of using the pre image?
This is an idea for your use: