Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 nW \subset \mathbb{R}^n” to “W n+1W \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 1\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 σ 1\sigma^{-1} manifestly continuous? You’ve already expressed r 2r^2 as a continuous function of the y iy_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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)