Map Projections #8

Zoë

Use the Jacobian method to compute the area distortion of general cylindrical projection and show that you obtain the same result we derived using just scaling factors.

Red

Given a map projection $T$, the area of distortion of $T$ from the globe to the map is $sec(\phi)JT$. The general cylindrical projection is denoted as $T(\phi,\theta)=(\theta,h(\phi))$.

Thus,
$JT=\begin{pmatrix} x_\theta &x_\phi\
y_\theta&y_\phi\end{pmatrix}=
\begin{vmatrix} 1 &0 \
0&h'(\phi) \end{vmatrix}=h'(\phi)$

|$x_\theta$ $x_\phi$| =|1 ___ 0|

|$y_\theta$ $y_\phi$| =|0 $h'(\phi)$|= $h'(\phi)$

(I can't get matrices to work properly)

So the area of distortion is $sec(\phi)h'(\phi)$

Now, the area of distortion can also be computed using scaling factors such that $distortion=M_\phi\dot M_\theta$. For cylindrical projections, $M_\phi=sec(\phi)$ and $M_\theta=h'(\phi)$ so $M_\phi\dot M_\theta=sec(\phi)h'(\phi)$, thus we have obtained the same result.