Definition:Stereographic Projection

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Definition

Let $\Bbb S$ be a sphere which is to be projected onto a plane.

Let $N$ be a distinguished point of a sphere $\Bbb S$.

Let $\PP$ be a plane perpendicular to the diameter of $\mathbb S$ through $N$.


Let $A$ be an arbitrary point on $\PP$.

Let the line $NA$ be constructed.

Spherical-Representation-of-Complex-Number.png

Then $NA$ passes through a point $A'$ of $\mathbb S$.

Hence $A'$ on $\Bbb S$ can be identified with $A$ on $\PP$.


Thus any point on $\PP$ can be represented by a point on $\mathbb S$.

This is known as a stereographic projection of $\mathbb S$ onto $\PP$.


With this construction, the point $N$ on $\mathbb S$ maps to no point on $\mathbb S$.



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Pole

The distinguished point $N$ is known as the pole of the stereographic projection.


Also presented as

In their account of stereographic projection, some sources are specific about where the plane is located with respect to the sphere.

For example: tangent to $\Bbb S$ at the opposite end of the diameter through $N$.

However, technically it does not matter where the plane is located, as long as it does not pass through $N$ itself.


Also see

  • Results about stereographic projections can be found here.


Sources

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