Parametrization of a Spherical Cap

Question

I'm looking at a source that says an upper cap of a sphere has parametrization limits...

$0 \le \theta \le 2\pi$

$0 \le \phi \le \pi/3$

I know the full half-sphere would be $0 \le \phi \le \pi$.

But if it's the upper cap, wouldn't it be $\pi/3 \le \phi \le \pi$?

Answer 1

This depends on your convention for the parameter $\phi$. The standard convention is that this angle (called the azimuthal angle) denotes the angle between a vector in $\mathbb{R}^3$ based at $0$ and the positive $z-$axis. In particular, $\phi=0$ gives the ray consisting of the positive $z-$axis.

So, if your sphere has radius $1$, then the spherical cap in question is given by $\rho =1$, $0\le \phi\le \frac{\pi}{3}$ and $0\le \theta \le 2\pi$.

What your parametrization with $\frac{\pi}{3}\le \phi\le \pi$ parametrizes is the lower "cap" of the sphere.