Let $X(\theta,\phi)=(\sin \theta \cos \phi, \sin\theta\sin \phi, \cos\theta)$ be parametrization of the sphere $S^2$. Let $P$ be the plane $x=z \cot\alpha$, $0<\alpha<\pi$ and $\beta$ be the angle wich the curve $P\cap S^2$ makes with the semimeridian $\phi=\phi_0$. Compute $\cos \beta$.
This is question 2 of sec. 2.5 of DoCarmo – Differential Geometry of curves and surfaces. Which is supposed to use concepts concerning the First Fundamental Form.
I finded a parametrization of the intersection curve as follows:
if $x=z\cot\alpha$, then
$z^2\cot^{2}\alpha+y^2+z^2=1$ which implies that $y^{2}+\dfrac{z^{2}}{\sin^{2}\alpha}=1$.
So, if $y(t)=\cos t$, $z(t)=\sin\alpha\sin t$, for the condition of intersection we will have $x(t)=\cos\alpha\sin t$. So $\lambda(t)=(x(t),y(t),z(t))$ is a parametrization of the intersection. But from now on I could not do anything to solve the problem. I think I must to write $\lambda^\prime$ in terms of the basis of the tangent vector of $S^2$. But this gave me a system that I couldn't solve. Any hint?.
Remark: I know that there is some topics in this site about this question. But I couldn't understand the answers.
Best Answer
HINT: You should work with the parametric equations of the sphere from the beginning. We have $x=\sin\theta\cos\phi = (\cot\alpha) z = \cot\alpha\cos\theta$, so $\tan\theta\cos\phi=\cot\alpha$. Can you find the tangent vector at the point where $\phi=\phi_0$?