spherical.to.polar.area: Convert latitude on sphere to radial variable in...

In retistruct: Retinal Reconstruction Program

Convert latitude 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)

Arguments

phi	Latitude
R	Radius

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.

Value

Coordinate rho that has the dimensions of length

Author(s)

David Sterratt

retistruct documentation built on April 3, 2025, 5:58 p.m.