Cylindrical Equal-Area Projection
April 10, 2023In 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 publication 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.
| Coefficient | Clarke 1866 | WGS84 |
|---|
| $b$ | 0.9991507116 | 0.9991600674 |
| $a_2$ | -0.0008471546 | -0.0008378456 |
| $a_4$ | 0.0000021283 | 0.0000020818 |
| $a_6$ | -0.0000000054 | -0.0000000052 |
| $b_2$ | -0.0001412092 | -0.0001396573 |
| $a_{22}$ | -0.0001411259 | -0.0001395758 |
| $a_{24}$ | 0.0000000839 | 0.0000000821 |
| $a_{26}$ | 0.0000000006 | 0.0000000006 |
| $b_4$ | -0.0000000435 | -0.0000000425 |
| $a_{42}$ | -0.0000000579 | -0.0000000567 |
| $a_{44}$ | -0.0000000144 | -0.0000000141 |
| $a_{46}$ | 0.0000000000 | 0.0000000000 |