spherical.to.polar.area: Convert lattitude on sphere to radial variable in
area-preserving projection
Description
Project spherical coordinate system $(\phi, \lambda)$ to a polar
coordinate system $(\rho, \lambda)$ such that the area of each
small region is preserved.Usage
spherical.to.polar.area(phi, R = 1)
Value
- Coordinate
rho that has the dimensions of length
Details
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.