How Do You Solve a Stereographic Projection Problem in Mathematics?

[Mathman23]

Urgent Stereographic projection question...

Homework Statement

Given the unit sphere [tex]S^2= \{x^2 + y^2 + (z-1)^2 = 1\}[/tex]

Where N is the Northpole = (0.0.2) the stereographic projection

[tex]\pi: S^2 \sim \{N\} \rightarrow \mathbb{R}^2[/tex] carries a point p of the sphere minus the north pole N onto the intersection of the xy plane with the straight line which N to p. Then if [tex](u,v) = \pi(x,y,z)[/tex] where [tex](x,y,z) \in S^2 \sim \{N\}[/tex] and [tex](u,v) \in xy [/tex] plane.

(1)Then I need to show the following:

[tex] \pi^{-1}: \mathbb{R}^2 \rightarrow S^2 [/tex]

[tex]\pi^{-1} \Biggr( \begin{array}{ccc}x = \frac{4u}{u^2+v^2+4} \\ y = \frac{4v}{u^2+v^2+4} \\ z =
\frac{2(u^2+v^2)}{u^2+v^2+4} \end{array}[/tex]

(2)Show that this can be covered by two coordinant neighbourhoods using stereographic projection.

Homework Equations

Could somebody please give me a small hint maybe or how to proceed here??

The Attempt at a Solution

(1) Here I need to find the unique line from N to xy-plane, which I get to

x = ut, y = vt and z = 2-2t. For t = 0; (x,y,z) = (0,0,2) and for t = 1; (x,y,z) = (u,v,0)

This unique line intersects with the sphere at:

[tex](ut)^2 + (vt)^2 + (1-2t)^2 = 1 \Rightarrow (4+u^2 + v^4) \cdot t^2 - 4t =0 [/tex]

where we interested at the solution [tex]t = \frac{4}{4+u^2+v^2}[/tex]

Thus the point of intersection with the sphere is

[tex]x = \frac{4u}{u^2+v^2+4}[/tex] and [tex]y = \frac{4v}{u^2+v^2+4}[/tex] and [tex]z = \frac{2(u^2+v^2)}{u^2+v^2+4}[/tex]

[tex]\pi_{1}(u,v) = \frac{4u}{u^2+v^2+4},-\frac{4v}{u^2+v^2+4}, \frac{2(u^2+v^2)}{u^2+u^2+4}[/tex](2)

If let's say a parameterzation around the North Pole by sending (u,v) to (u,-v,0), the resulting map

[tex]\pi_{2}(s,t) = \frac{4s}{s^2+t^2+4},-\frac{4t}{s^2+t^2+4}, \frac{2(s^2+t^2)}{s^2+t^2+4}[/tex]

it sends the unit circle to the equator and the unit disk to the Northern hemisphere.

It covers the sphere except for the Southpole which is covered by the parameterization in (1). Ergo these two maps or neighbourhood cover the entire sphere.

By simple Algebra

[tex]\pi_{2}(u,v) = \pi_{1}(\frac{4s}{s^2 + t^2}, \frac{-4t}{s^2 +t^2})[/tex]

[tex]\pi_{1}(s,t) = \pi_{2}(\frac{4u}{u^2 + v^2}, \frac{-4v}{u^2 +v^2})[/tex]Q.e.d.

Is this correct?? Or am I on the wrong track?? I hope somebody would answer cause you guys are my only hope :|
Best Regards
Fred

 

[mighty2000]

Thank you for your question. It seems like you have made some good progress in solving the problem. I would suggest double-checking your calculations and making sure that the parametrizations you have found actually cover the entire sphere. You can do this by plotting the parametrizations and seeing if they cover all points on the sphere.

Additionally, you may want to explain your reasoning and steps more clearly in your solution. This will make it easier for others to understand and verify your solution. Also, don't be afraid to ask for help from your peers or your instructor if you are stuck. Sometimes discussing the problem with others can help you see things from a different perspective and come up with a solution.

Best of luck with your problem!