I have some pairs of real numbers $(\rho_1,\alpha_1),\dots (\rho_n, \alpha_n)$. I know that all my $\rho$'s are positive, but there is no constraints on my $\alpha$'s. I want to find a function $\phi$ such as $((\rho_1,\theta_1),\dots,(\rho_n,\theta_n)$ are some cartesian products, with $\theta_i = \phi(\rho_i,\alpha_i)$.
Is there a way to find such a $\phi$? Thanks!