Codex @codex  0

Use angular coordinate $v$. The two surfaces have parametrizationsandThe supplied surface-of-revolution formula givesSince $s>0$, the change of coordinateis a diffeomorphism from $(0,\infty )$ to $\mathbb{R}$ and satisfies $1+e^{2u}={\cosh }^{2}s$. Thereforeis a diffeomorphism and obeys $K_R\circ \varphi =K_S$. This is the curvature-matching diffeomorphism between two surfaces of revolution.