I am really new in geometry and especially in working with stereographic projection, so excuse me, please, if my question is too easy.
Given is the ellipsoid: $E = \left \{ (x,y,z)\in \mathbb{R}^{3}: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} +\frac{z^{2}}{c^2} = 1\right \}$.
I have to find two parametrizations with the following points excluded: $E\setminus\{(a,0,0)\}$ and $E\setminus\{(0,0,-a)\}$.
OK, we know the definition of the stereographic projection of the unit sphere in $\mathbb{R}^3$ with excluding the north pole $(0,0,1)$. It is given by: $(x,y,z)=\left ( \frac{2x}{x^{2}+y^{2}+1}, \frac{2y}{x^{2}+y^{2}+1}, \frac{x^{2}+y^{2}-1}{x^{2}+y^{2}+1} \right )$.
I know, the problem takes much time to do the calculations, so i would be very glad if someone could give a hint how to do this calculation, because i really don't get it. Thank you very much in advance.
Best Answer
A hint for the first one:
Any point $P:=(0,u,v)$ in the $(y,z)$-plane determines a line $g_P:=P\vee A$, where $A:=(a,0,0)$. Intersecting $$g_P:\quad t\mapsto \bigl((1-t) a,t u,t v\bigr)\qquad(-\infty<t<\infty)$$ with the ellipsoid $E$ you get a quadratic equation for $t$ with one obvious solution $t=0$. The other solution leads you to the point $(x_P,y_P,z_P)\in E$ stereographically related to $P$. All in all you will obtain a parametric representation $$(u,v)\mapsto\bigl(x_{(u,v)},y_{(u,v)},z_{(u,v)}\bigr)\in E\setminus\{(a,0,0)\}\ .$$