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This item got one answer after some hours on stackexchange, and I have a feeling I should solicit whatever variety of opinions may be out there:

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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When I learned to talk mathematics, "stereographic projection" or "the inverse of stereographic projection" was the standard terminology that I learned for all of the cases you mention, regardless of the dimension of the domain and/or range. I guess it's such a useful and evocative terminology that the restriction of meaning you mention has been lost. – Lee Mosher Jan 24 '13 at 0:48
I've got to agree with Lee; after all, nobody seems to worry too much about the use of the word `volume' to describe measure in n dimensions for $n>3$. – Danny Ruberman Jan 24 '13 at 1:32
I wouldn't be ultra-fussy in hewing to word origins. Mathematicians are constantly reappropriating words for their purposes, and one of these purposes is understanding the power of generalization. – Todd Trimble Jan 24 '13 at 1:42
I would shamelessly use "stereographic projection" even for the circle. And call the inverse map "inverse stereographic projection". Anything else just complicates things. – Deane Yang Jan 24 '13 at 2:49
Overreliance on word-origins can lead one astray. My little dictionary of classical Greek gives “income; profit; gain; gratification” as the meaning of $\lambda\widetilde\eta\mu\mu\alpha$. – Lubin Jan 24 '13 at 15:07
up vote 1 down vote accepted

My standard reference for elementary geometry is the book M. Berger, Geometry. In section 18.1.4 he defines ``stereographic projection'' in any dimension. Of course it was originally introduced for 2-dimensional sphere, and the name comes from this original use. But nowadays this term is used in any dimension, and I do not see why dimension 1 must be an exception. For the inverse map, Berger has no name, just calls it the inverse stereographic projection.

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