# maps inter surfaces

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

Filed under math

### 3 responses to “maps inter surfaces”

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 $S^2$