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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?

Michael Hardy
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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 00:48
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    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 01:32
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    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 01:42
  • @DannyRuberman : My qualms were about $n<3$, not about $n>3$. I've done some editing of the question to make that even more obvious than it already was. – Michael Hardy Jan 24 '13 at 02:36
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    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 02:49
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    Danny could just as well have written $n\neq 3$ instead of $n>3$. – Tom Goodwillie Jan 24 '13 at 04:34
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    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
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    Totally agree with Deane (and not only "would I", but "I do", as when using stereographic projection from the circle to the line as a means of characterizing Pythagorean triples). – Todd Trimble Jan 24 '13 at 15:55
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    Variations of meaning from the original etymology, sometimes even to the opposite, is a natural and fashinating phenomenon of language. It's interesting and useful to know the origin and the history of a word, but that doesn't mean one has to use it in the pristine sense. So I personally use "stereographic projection" in any dimension, and I'm glad to know of that small abuse of language for n=1. – Pietro Majer Jan 24 '13 at 21:04

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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.

Alexandre Eremenko
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