Solid angle subtended in latitude-longitude maps

Question

I need to scale a latitude-longitude map with the solid-angle each "pixel" subtend. How can I obtain the said solid angle starting from the $\phi$ and $\theta$ angles?

Thank you very much

Answer 1

If $\theta=$ angle of longitude & $\phi=$ angle of latitude then by using double integration, we can calculate solid angle as follows $$\Omega=\iint_S\sin\theta d\theta d\phi$$ Now, if an symmetrical area on a spherical surface subtends an angle of logitude $\theta$ & an angle of latitude $\phi$ at the center of the sphere then the solid angle subtended by the (symmetrical) area is given as $$\Omega=\int_{0}^{\phi}d\phi\int_{0}^{\theta}\sin\theta d\theta$$ $$=\int_{0}^{\phi}d\phi [-\cos\theta]_{0}^{\theta}$$ $$=\int_{0}^{\phi}d\phi [1-\cos\theta]=[1-\cos\theta][\phi]=\phi(1-\cos\theta)$$
Above formula is valid only for a symmetrical area subtending angles $\phi$ & $\theta$ at the center of a spherical surface.