Project spherical coordinate system \((\phi, \lambda)\) to a polar coordinate system \((\rho, \lambda)\) such that the area of each small region is preserved.
spherical.to.polar.area(phi, R = 1)Latitude
Radius
Coordinate rho that has the dimensions of length
This requires $$R^2\delta\phi\cos\phi\delta\lambda = \rho\delta\rho\delta\lambda$$. Hence $$R^2\int^{\phi}_{-\pi/2} \cos\phi' d\phi' = \int_0^{\rho} \rho' d\rho'$$. Solving gives \(\rho^2/2=R^2(\sin\phi+1)\) and hence $$\rho=R\sqrt{2(\sin\phi+1)}$$.
As a check, consider that total area needs to be preserved. If \(\rho_0\) is maximum value of new variable then \(A=2\pi R^2(\sin(\phi_0)+1)=\pi\rho_0^2\). So \(\rho_0=R\sqrt{2(\sin\phi_0+1)}\), which agrees with the formula above.