Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Finally created stereographic projection. Lots more could be said, including metric/conformal aspects.
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.
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.
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.
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.
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.
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. :-)
Thanks, it must have been getting too late for me last night…
I added a bit to stereographic projection, speaking about geometry over other fields, and giving Pythagorean triples as an example.
1 to 9 of 9