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Problem 5

# Show that a circle on the sphere that does not pass through the north pole corresponds to a circle in the complex plane.

Expert verified
Stereographic projection is used to establish the correspondence between a circle on a sphere not passing through the North pole, and a circle in the complex plane. Key to this proof is the invariant angle property of the stereographic projection.
See the step by step solution

## Step 1 - Construct the stereographic projection

A circle on the sphere can be stereographically projected onto the complex plane. Let's take the north pole of the sphere as $$N$$ and let $$O$$ be a point on the circle on the sphere that doesn't pass through $$N$$. Draw a line from $$N$$ through $$O$$ to the complex plane. The point where it intersects the plane is the stereographic projection corresponding to $$O$$.

## Step 2 - Establish connection between points

The key point here is to realize that all points $$O$$ on the circle (which do not pass through north pole) will maintain the same angle with respect to $$N$$. Hence, the radii from the origin of the complex plane to projected points are all at the same angle. This automatically gives us a circle in the complex plane.

## Step 3 - Define the circle in the complex plane

Keeping the above properties in mind, define a circle in the complex plane that mirrors the circular structure in spherical coordinates. We can state that for any point $$O$$ on the circle on the sphere, its projection will always lie on the defined circle in the complex plane, hence proving the correspondence between them.

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