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I'm wondering about the proper use of the term "stereographic projection".

Draw a line from a point on a sphere, which let us call the north pole, through another point on the sphere, to a plane parallel to the plane tangent to the sphere at the north pole. That last point is the stereographic projection of the typical point on the sphere onto that plane. Then the same thing gets done in higher dimensions and the same term --- "stereographic projection" --- is used.

No problem so far.

But I hesitate to use that term when it's from a circle to a line, because "ster-" or "stere-" usually means "solid" or "three-dimensional".

Are there opinions on the propriety of that usage?

Also, is there a name for the inverse mapping from the line or plane or hyperplane to the sphere?

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up vote 1 down vote accepted

If stereographical projection is taken as a homeomorphism between the whole space $R^n$ and unit sphere without the pole $S\subset R^{n+1}$, then dimension should not change the terminology.

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There is certainly no homeomorphism between $\mathbb R^{n+1}$ and any subset of $\mathbb R^n$. – Michael Hardy Jan 24 '13 at 2:44
sorry for the nonsensical statement – mez Jan 25 '13 at 12:14

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