Suppose you have a Christmas ball, which has a decorative lining around it so that it forms a spherical spiral around it. I want to find how long that decorative lining can be given some parameters. This is more or less the shape I am looking at. This was taken from Wikipedia at this link.
The parametric equations of this were also provided:
$$x = r \cdot \sin(\theta) \cdot \cos(c\theta)$$ $$y = r \cdot \sin(\theta) \cdot \sin(c\theta)$$ $$z = r \cdot \cos(\theta)$$ Where $0 \le \theta \le \pi$.
$c$ is twice the number of turns, and in my specific case, I set $c=8$, just like in the picture, for simplicity sakes. $r$ is the radius, and in my specific case the radius is $r=4cm$. What I am looking for is the length of the red line. How do I find it?
I'm going to use the parameter $t$ because I'm more comfortable with it. Our parametric equations are: $$\begin{bmatrix} x( t)\\ y( t)\\ z( t) \end{bmatrix} =r\begin{bmatrix} \sin( t)\cos( ct)\\ \sin( t)\sin( ct)\\ \cos( t) \end{bmatrix}$$ The formula for the arc length is $$s(t)=\int_0^t \sqrt{\dot{x}(t')^2+\dot{y}(t')^2+\dot{z}(t')^2}~\mathrm{d}t'$$ We can compute $$\begin{bmatrix} \dot{x}( t)\\ \dot{y}( t)\\ \dot{z}( t) \end{bmatrix} =r\begin{bmatrix} \cos( t)\cos( ct) -c\sin( ct)\sin( t)\\ \cos( t)\sin( ct) +c\cos( ct)\sin( t)\\ -\sin( t) \end{bmatrix}$$ Now I hope you can trust me when I say that $\dot{x}(t)^2+\dot{y}(t)^2+\dot{z}(t)^2=r^2(c^2\sin^2(t)+1).$ (You can verify the algebra yourself if you want.) Therefore, taking the endpoint of integration to be $\pi$, our integral is $$s(\pi)=\int_0^\pi r\sqrt{c^2\sin^2(t)+1}~\mathrm{d}t$$ Unfortunately this integral doesn't have any nice expressions in terms of elementary functions, but we can write it as $$s(\pi)=r\left(E(-c^2)+\sqrt{c^2+1}E\left(\frac{c^2}{c^2+1}\right)\right)$$ Where $E$ denotes the complete elliptic integral of the second kind. You can evaluate this numerically for different values of $c$ and $r$ if you like.