Circles to Lines in the Stereographic Projection of a Sphere

Question

I've been learning about stereographic projections of spheres recently, and there were two things I couldn't intuitively understand...

My first question was why it was that the projection preserved circles on the sphere. However, after watching this video here...

https://www.youtube.com/watch?v=QqxNpYQYTFA

...I now understand that.

My second question is why it is that a circle on the sphere that passes through the point where the light rays emanate from gets projected onto a straight line? I still can't wrap my head around it...although I feel like it should seem obvious?

Thanks!

Answer 1

I don't want to take credit for this answer, but in case anyone is interested for an intuitive explanation:

https://www.reddit.com/r/3Blue1Brown/comments/eb6bta/on_some_properties_of_stereographic_projection/

Instead of thinking of the circle, think of a plane that cuts through the horizontal plane that's being projected onto, and also cuts through the point of projection.

Then intersection of the two planes will be a line. That is, every point on the plane that cuts through the horizontal plane and through the point of projection will be projected onto that line.

Any circle on the sphere is the intersection of that plane and the sphere. Therefor, that circle gets projected to that line as well.