I've been practicing drawing surfaces in different coordinates. I can do the easier ones but no I am completely stuck on the following two:
Say we define spherical coordinates as follows:
so that $\theta \in [0,2\pi]$ and $\phi \in [0,\pi]$.
Then I want to draw (without using software):
$$ \rho = \phi$$
and
$$ \rho = \cos \phi$$
Here is how far I got so far:
Luckily, the angle $\theta$ is free so all we have to do is find the shape in the $xz$-plane and then rotate. So this reduces to drawing in polar coordinates.
Unfortunately, this insight does not help me further since I am equally stuck in only 2 dimensions when it comes to polar coordinates.
For the first one I see that as the angle grows so does the radius but this doesn't help me draw the exact shape. It crosses the $x$ axis at $\pi$ and the $z$ axis at $-2\pi$ but again this doesn't give me the exact shape.
For the other equation I tried to rewrite it:
$\cos \phi = {y\over \rho}$
but I do not see how this will help me draw the function.
Please could someone help me and explain to me how to draw such functions?