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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.?

Mika H.
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    A circle on the sphere is the intersection of the sphere with a plane (such that the intersection does not consist of a single point). – Daniel Fischer Sep 01 '13 at 21:23
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    A circle on the sphere is literally a circle: it is the set of all points in a plane at a specified distance from one point in the plane. There is also a point on the sphere, not on the plane from which every point on the circle is equidistant regardless of whether on measure distances along the surface or along straight lines through the interior. – Michael Hardy Sep 01 '13 at 23:28

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Here is an interactive demonstration on how to visualise it.

Play with it until you get the hang of it. It's actually quite fun.

You can imagine a unit sphere sitting on the complex plane. A line on the complex plane, can be though of as a plane $X$'s intersection through the complex plane. $X$ is oriented such that it has to touch the top of our sphere. The intersection between $X$ and our sphere is the corresponding circle on the sphere.

A circle on the complex plane works the same way. Let us call a circle on the complex plane $Y$. All the points on $Y$ are connected the the top point of our sphere creating a curved plane that intersects our sphere and makes a circle. Perhaps this is also anther way of thinking of the line (and further shapes!).

A line being mapped to a circle:

enter image description here

A circle being mapped to a circle:

enter image description here

Robert Frost
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Ali Caglayan
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    Since stereographical projection maps circles in circles, given a circle $B={|z|<r}$ in $\overline{\Bbb C}$, is there a known formula which expresses the radius of the image of $B$ (which is again a circle) inside $\Bbb S^2$? What about ${|z|>r}$? – Joe Jul 27 '20 at 16:49