Serendipitous Infinity Symbol

I accidentally found a way to generate a curve that looks like an infinity symbol when projected onto the $xy$ plane.

$\large{\theta = cos \, \alpha \\ \phi = sin \, \alpha \\ 0 \leq \alpha \leq 2 \pi \\ x = cos \theta \, sin \phi \\ y = sin \theta \, sin \phi \\ z = cos \phi}$

What is the length of the curve in $xyz$ coordinates, when plotted over the full range of $\alpha$ ?

Note: We're interested in the length of the full 3D curve; not the projection onto the $xy$ plane