I was wondering what would be an equation for Hadamard operation for a single qubit, given the input as the current $\theta$ (0 to $+\pi/2$) and $\varphi$ ($-\pi$ to $+\pi$) and output expected in $\theta$ and $\varphi$ with same ranges. Most expressions of Hadamard that I saw use Cartesian transformation, but not $\theta/\varphi$ transform.
I could convert the input to Cartesian form, and convert the output back into $\theta/\varphi$ form well, but I'm looking for an equation that does it without using Cartesian conversion step? The goal is to understand direct relation between input and output. I tried interpreting Hadamard as subtracting $\pi/4$ from $\theta$ and adding $\pi$ to phase, but I see that it doesn't quite work for arbitrary input.
Note: here $\varphi$ refers to relative phase ($-\pi$ to $+\pi$) and $\theta$ refers to component amplitude factor (0 to $+\pi/2$).