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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?

Andres Mejia
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combo student
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3 Answers3

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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,

$$x=at$$

$$y=bt$$

$$z=1-t$$

Ahmed S. Attaalla
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  • Thank you, follow up question: To find the intersection with the sphere, we must compute when the right hand side has norm 1, why is that? – combo student Mar 07 '17 at 04:34
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    @combostudent: What does "intersect with $S^2$" mean? – Ted Shifrin Mar 07 '17 at 04:41
  • To find the insertion we must have $x^2+y^2+z^2=1$ (we must be on the sphere) and $x=at$, $y=bt$, $z=1-t$ (we must also be on the line). Putting this together gives $(at)^2+(bt)^2+(1-t)^2=1$. @combostudent Geometrically, we are looking for the point(s) $(at,bt,1-t)$ a unit away from the origin so that we are on the sphere. – Ahmed S. Attaalla Mar 07 '17 at 04:46
  • I think the op means intersection of the sphere and line. Is that what you mean? @combostudent – Ahmed S. Attaalla Mar 07 '17 at 04:52
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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

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

Mark Viola
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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.

Andres Mejia
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