maps inter surfaces


carrying patterns on a surface (Monge's like) to the 2-sphere

3 Comments

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3 responses to “maps inter surfaces

  1. Amadeus

    Is this for every surface sigma, even if it isn´t an sphere?

    • k-qit

      it only works when we are projecting on the sphere and the surface \Sigma is described Monge’s like, cuz the stereo-graphic projection is too particular…

      Perhaps the next case is to carry patterns among two surfaces described Monge’s like each…

  2. k-qit

    the stereographic-projection is

    \left(\begin{array}{c}v\\w\end{array}\right)\stackrel{\phi}\mapsto\left(\begin{array}{c}2v/(1+v^2+w^2)\\2w/(1+v^2+w^2)\\(v^2+w^2-1)/(1+v^2+w^2)\end{array}\right)

    which carries points in \mathbb{R}^2 to the 2-sphere.

    Points in the surface \Sigma are of the form p=\left(\begin{array}{c}a\\b\\f(a,b)\end{array}\right) for a differentiable function f:\Omega\to\mathbb{R}.

    Then to carry the point p\in\Sigma to S^2 just apply \phi to the pair \left(\begin{array}{c}a\\b\end{array}\right).

    In this way you gonna get for a finite set of point in \Sigma another finite set in S^2

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