 Solution

Use the convention $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$, with $\theta$ measured from the positive vertical axis. The squared radius isSince the horizontal radial distance is $1/\sqrt{1+t^2}>0$ for finite $t$,The horizontal angle advances with $t$. Thus the spherical coordinates areAn unwrapped continuous azimuth is simply $\varphi =t$; a principal-angle convention introduces harmless jumps by $2\pi$. The polar angle stays in $(0,\pi )$ and approaches the two poles only as $t\to \pm \infty$.