How to represent this parametrically?

Question

For the purpose of solving a problem involving manifolds, I want to know how to represent this situation...

I have an $S^2$ sphere $(x^2 +y^2 + z^2 =1)$ and a point $(a,b)$ in the plane $\mathbb{R}^2$. I want to connect it with the north pole of the sphere $(0,0,1)$ by a line to compute its intersection with $S^2$.

How do I find the equation of this line?

Answer 1

The line goes through the points:

$$P_1(0,0,1)$$

$$P_2(a,b,0)$$

The direction vector, from $P_1$ to $P_2$ is, $\langle a-0,b-0,0-1 \rangle$. This is $\langle a,b,-1 \rangle$. Then the equation of the line is given by the position vector function,

$$\vec r(t)=\langle a,b,-1 \rangle t+\langle 0,0,1 \rangle$$

So we may parametrize as follows,

$$x=at$$

$$y=bt$$

$$z=1-t$$

Answer 2

The vector from the point $(a,b,0)$ to the point $(0,0,1)$ can be written

$$\vec V=(a,b,-1)$$

and the equation of the line passing through $(a,b,0)$ and $(0,0,1)$ can be expressed parametrically by

$$\vec r(s)=(0,0,1)+\vec V\,s$$

Answer 3

I hope it is clear that any such map will not cover the whole sphere, since the chart is not defined at the north pole. On the other hand, consider the map

$$(x,y) \mapsto \frac{1}{1+x^2+y^2}(2x,2y,x^2+y^2-1).$$

I think that this is what you are looking for, since it tells you where the line intersects the sphere, and gives you local co-ordinates for each point on the sphere. You should check to see why this is the correct map, and also find its inverse as an exercise.