How to Make Circles Into Circles on a Different Shape

Elkement, who’s been a longtime support of my blogging here, has been thinking about stereographic projection recently. This comes from playing with complex-valued numbers. It’s hard to start thinking about something like “what is 1 \div \left(2 + 3\imath \right) and not get into the projection. The projection itself Elkement describes a bit in this post, from early in August. It’s one of the ways to try to match the points on a sphere to the points on the entire, infinite plane. One common way to imagine it, and to draw it, is to imagine setting the sphere on the plane. Imagine sitting on the top of the sphere. Draw the line connecting the top of the sphere with whatever point you find interesting on the sphere, and then extend that line until it intersects the plane. Match your point on the sphere with that point on the plane. You can use this to trace out shapes on the sphere and find their matching shapes on the plane.

This distorts the shapes, as you’d expect. Well, the sphere has a finite area, the plane an infinite one. We can’t possibly preserve the areas of shapes in this transformation. But this transformation does something amazing that offends students when they first encounter it. It preserves circles: a circle on the original sphere becomes a circle on the plane, and vice-versa. I know, you want it to turn something into ellipses, at least. She takes a turn at thinking out reasons why this should be reasonable. There are abundant proofs of this, but it helps the intuition to see different ways to make the argument. And to have rough proofs, that outline the argument you mean to make. We need rigorous proofs, yes, but a good picture that makes the case convincing helps a good deal.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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