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

In retistruct: Retinal Reconstruction Program

Description

Project spherical coordinate system (φ, λ) to a polar coordinate system (ρ, λ) 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^2δφ\cosφδλ = ρδρδλ

. Hence

R^2\int^{φ}_{-π/2} \cosφ' dφ' = \int_0^{ρ} ρ' dρ'

. Solving gives ρ^2/2=R^2(\sinφ+1) and hence

ρ=R√{2(\sinφ+1)}

.

As a check, consider that total area needs to be preserved. If ρ_0 is maximum value of new variable then A=2π R^2(\sin(φ_0)+1)=πρ_0^2. So ρ_0=R√{2(\sinφ_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 4, 2020, 5:08 p.m.