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• CommentRowNumber1.
• CommentAuthorDavidRoberts
• CommentTimeDec 10th 2015

Finally created stereographic projection. Lots more could be said, including metric/conformal aspects.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 10th 2015

Good that you created this.

Just for the record, I made two trivial edits: changed “$W \subset \mathbb{R}^n$” to “$W \subset \mathbb{R}^{n+1}$” and changed “isomorphism” to “homeomorphism”.

Also at representation sphere – construction I have edited ever so slightly, trying to clarify a bit more.

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeDec 10th 2015

I added a little on preserved structures. Could obviously be expanded. I intend to put the inverse formula, and disciss special cases at some point.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 7th 2017
• (edited May 7th 2017)

I have expanded at stereographic projection for the purpose of exposition: I gave the entry an actual Idea-section and then I wrote out the argument for why the projection is indeed a homeomorphism in some pedantic detail.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeMay 7th 2017

I didn’t know we had so little there. I often use the phrase “stereographic projection” more generally, e.g., in setting up isomorphisms between conic sections and $\mathbb{P}^1$ over more general fields.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMay 7th 2017

Yes, the entry should be expanded. But right now I just needed it to contain details on the classical situation. I am still looking for a non-cumbersome formal argument for the last sentence here.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeMay 7th 2017
• (edited May 7th 2017)

Isn’t the inverse function $\sigma^{-1}$ manifestly continuous? You’ve already expressed $r^2$ as a continuous function of the $y_i$, etc., so it seems to me all your work is done. :-)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 8th 2017

Thanks, it must have been getting too late for me last night…

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeMay 8th 2017

I added a bit to stereographic projection, speaking about geometry over other fields, and giving Pythagorean triples as an example.

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