So I'm plotting curves in three dimensions on my computer. I'm currently using equations in spherical coordinates to define them. These are some examples, using ISO convention ( r , θ , φ ):
$\big(a.sin(b.\theta), \theta, c\big)$
$\big(a.sin(b.\theta), \theta, \theta\big)$
$\big(a.abs(sin(b.\theta)), \theta, c.sin(d.\theta)\big)$
How could I find the closing angle for these variations?
Maybe a good starting point should be looking at 2D rhodoneas. According to wolfram, a two-dimensional rhodonea curve ($r = a.cos(\frac{r}{s}.\theta)$) closes at a polar angle of $\theta=\pi .s.p$, where $p=1$ if $r.s$ is odd and $p=2$ if $r.s$ is even. But I can't understand why that happens.
Ref: https://mathworld.wolfram.com/Rose.html
I'm probably missing something or maybe I got something wrong, so if you have any thoughts on this I would love to know and discuss it further.
If i understand correctly you are asking for the angle at which the curves are 'back at their starting point'. Let's start with the rhodoneas. You might know that the $\sin$ and $\cos$ functions are periodic with a period of $2\pi$, meaning that they are back at their starting point at an angle of $2\pi$ radians. A curve of the form $r=a\cos(\frac{n}{s} \theta)$ is back at its starting point when $\frac{n}{s} \theta$ is a multiple of $2\pi$ for the first time. If $n$ is even (a multiple of 2) this happens at $\theta=s \cdot\pi$, because $s \cdot \pi \cdot \frac{n}{s} = n\pi$ and $n$ is a multiple of two.
If $n$ is not even this happens at $\theta=2s \cdot\pi$.
Back to your curves:
$(a \sin(b\theta), \theta, c)$ closes when $b\theta$ and $\theta$ are both a multiple of $2\pi$ for the first time. If $b$ is an integer this happens at $\theta=2\pi$, if $b$ is a rational number given by $b=\frac{p}{q}$ with $p,q$ integers, this happens at $\theta=q\cdot 2\pi$.
If $b$ is irrational, the curve never closes because an integer multiple of $2\pi$ cant also be an integer multiple of $2\pi \cdot b$.
Similar logic applies to the other two curves.
EDIT:
I forgot to mention that $\theta$ needs to be an integer multiple of $2\pi$, otherwise the curve won't end at a point equivalent to $(0,0)$.