I'm reading about the spherical representation and the Riemann sphere, and the projection transformation that takes a point on the sphere to a point on the (extended) complex plane. The text says that, geometrically, the stereographic projection transforms every straight line in the complex plane into a circle on the sphere. Moreover, any circle on the sphere corresponds to a circle or straight line in the plane.
I'm having trouble visualizing this. How exactly is a circle on the sphere defined? Is it given by some kind of simple equation? And how do I see that a circle maps to a straight line, etc.?