I am trying to solve a problem in Do Carmo, a book which I often find incomprehensible.

Let $X(\varphi,\theta)=(\sin(\theta)\cos(\varphi),\,\sin(\theta)\sin(\varphi),\,\cos(\theta))$ be a parametrization of the unit sphere $S^2$. Let $P$ be the plane $x=z\cot(\alpha)$, $0<\alpha<\pi$, and $\beta$ be the acute angle which the curve $P\cap S^2$ makes with the semimeridian $\varphi=\varphi_0$. I am trying to compute $\cos(\beta)$.

I know I have to find a parametrization for both curves. First, how would I go about finding a parametrization $\gamma$ for $P\cap S^2$? The parametrization for the semimeridian I believe would be $\alpha(\varphi):=X(\varphi,\theta_0)$. Then once I have a parametrization for $P\cap S^2$ I would compute $$\cos(\beta)=\frac{\langle\alpha'(0),\,\gamma'(0)\rangle}{|\alpha'(0)|\,|\gamma'(0)|}.$$ Is this correct as well?


You should follow the calculation on p. 96 (of my edition, anyway), the loxodrome calculation, with one change. Instead of $\theta'$ and $\phi'$ being unknowns to be solved for, they are related by $\tan\theta\cos\phi = \cot \alpha$. Differentiate that to get a relationship between $\theta'$ and $\phi'$. Then you don't know what $\theta'$ and $\phi'$ are (because you don't have a parametrization of the curve) but you can determine the direction of the vector (i.e., up to a constant). That is good enough because we don't need unit vectors in the formula for the cosine of the angle between two vectors.

You also need to use the first fundamental form (i.e., E, F, and G) to calculate the length and inner product. It is similar to the loxodrome calculation on page 96.

After a bit of work I get the formula $\cos\beta = \sin\phi\sin\alpha$. I'm not absolutely sure it's correct but it passes a few basic tests. E.g., if $\phi=0$ or $\pi$, $\beta=\pi/2$ (the prime meridian cuts all these circles at right angles), and if $\phi=\pi/2$ then we have $\cos\beta=\sin\alpha$ which makes sense geometrically. ($\alpha$ and $\beta$ should be complementary angles in this case. I wish doCarmo would use colatitude instead of latitude in $x=z\cot\alpha$.)

This seems like an obvious exercise in spherical geometry, but I can't find verification of the result anywhere (which is what led me here). I'm going to use more general spherical trigonometry with the three great circles $x=z\cot\alpha$, $\phi=\phi_0$, and the prime meridian to verify the result. I think it will work.


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