Cylindrical Equal-Area Projection

# Cylindrical Equal-Area Projection

##### April 10, 2023

In his book Map Projections: A Working Manual, Chapter 10, Snyder gives a brief explanations of the formulas for oblique aspect of the ellipsoidal form of the Cylindrical Equal-Area projection. In short, Snyder gives a table of coefficients for a Fourier series used to calculate the projected coordinates. While it contains a brief section on how to calculate the coefficients for the Fourier series, I found it confusing. I had to dig another Snyder publication1 to understand how these are calculated.

The central point is the integration of the function:

\eqalign{ F =& \{\sin^2\beta_p\cos^2\phi_c/[(1-e^2\sin^2\phi_c)\cos^4\beta_c] \cr &+ (1-e^2\sin^2\phi_c)q_p\cos^2\beta_p\cos^2\lambda'/(4\cos^2\phi_c)\}^{1/2} } \tag{ 1 }
For the Fourier coefficients, the function is integrated once with respect to $\lambda’$ and then with respect to $\phi_p$, the latitude of the projection pole. In the end we end up with the following formulas:

\begin {align} q(\phi) &= (1-e^2)\left[\frac{\sin\phi}{1-e^2\sin^2\phi}-\frac{1}{2e}\ln\left(\frac{1-e\sin\phi}{1+e\sin\phi}\right)\right] \tag{1} \cr q_p &= q(\pi/2) \tag{2} \cr \beta(\phi) &= \arcsin(q(\phi)/q_p) \tag{3} \cr \beta_c(\phi_p, \lambda') &= \arcsin[\cos\beta(\phi_p)\sin(\lambda')] \tag{4} \cr \phi_c(\phi_p, \lambda') &= \phi(\beta_c(\phi_p, \lambda')) \tag{5} \end{align}
where $\phi(\beta)$, the function giving geodetic latitude fom authalic latitude, is an iterative function:
$$\phi_n = \phi_{n-1} + \frac{(1-e^2\sin\phi_{n-1})^2}{2\cos\phi_{n-1}}\left[ \frac{q_p\sin\beta}{1-e^2}-\frac{\sin\phi_{n-1}}{1-e^2\sin^2\phi_{n-1}}+\frac{1}{2e}\ln\left(\frac{1-e\sin\phi}{1+e\sin\phi}\right)\right] \tag{6}$$
with initial approximation: $\phi_0 = \arcsin[(q_p\sin\beta)/2]$.

We can rewrite $F$ as:

\eqalign{ F(\phi_p, \lambda') = \bigg [ & \frac{\sin^2\beta(\phi_p)\cos^2\phi_c(\phi_p,\lambda')}{(1-e^2\sin^2\phi_c(\phi_c, \lambda')\cos^4\beta_c(\phi_p,\lambda')} \cr &+ \frac{q_p^2(1-e^2\sin^2\phi_c(\phi_p, \lambda')\cos^2\beta(\phi_p)\cos^2\lambda'}{4\cos^2\phi_c(\phi_p,\lambda')} \bigg ]^{1/2} } \tag{ 7 }

A first integration with respect to $\lambda’$, gives coefficients that are functions of the pole latitude $\phi_p$:

\begin{align} B(\phi_p) &= \frac{2}{\pi}\int_0^{\pi/2}F(\phi_p, \lambda')\mathrm{d}\lambda' \cr \tag{8} \cr A_n(\phi_p) &= \frac{4}{\pi}\int_0^{\pi/2}F(\phi_p, \lambda')\cos(n\lambda')\mathrm{d}\lambda' \tag{9} \end{align}
A second integration with respect to $\phi_p$ gives Fourier coefficients that can be applied to any latitude:
\eqalign{ b &=\frac{2}{\pi}\int_0^{\pi/2}B(\phi_p)\mathrm{d}\phi_p \cr a_n &= \frac{4}{\pi}\int_0^{\pi/2}B(\phi_p)\cos(n\phi_p)\mathrm{d}\phi_p \cr b_n &= \frac{2}{\pi}\int_0^{\pi/2}A_n(\phi_p)\mathrm{d}\phi_p \cr a_{nm} &= \frac{4}{\pi}\int_0^{\pi/2}A_n(\phi_p)\cos(m\phi_p)\mathrm{d}\phi_p \cr }

These formulas can be used to calculate coefficients for any given ellipsoid. Bellow are the coefficients for Clarke 1866 and WGS84.

CoefficientClarke 1866WGS84
$b$0.99915071160.9991600674
$a_2$-0.0008471546-0.0008378456
$a_4$0.00000212830.0000020818
$a_6$-0.0000000054-0.0000000052
$b_2$-0.0001412092-0.0001396573
$a_{22}$-0.0001411259-0.0001395758
$a_{24}$0.00000008390.0000000821
$a_{26}$0.00000000060.0000000006
$b_4$-0.0000000435-0.0000000425
$a_{42}$-0.0000000579-0.0000000567
$a_{44}$-0.0000000144-0.0000000141
$a_{46}$0.00000000000.0000000000