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The Sinusoidal Projection is a coordinate transform often used to map geographical coordinates on to a flat surface in order to generate a map. It is typically stated that this projection is area-preserving, meaning that while shapes and angles may be distorted, the ratio of the area of any region calculated in the projected coordinates to the area measured on the surface of the globe will be constant with respect to position on the globe.

The equations relating projected coordinates to geographical coordinates are:

$$\begin{eqnarray} x &= &\lambda \cos{\phi} \\ y &= &\phi \end{eqnarray} $$

where $\lambda$ is longitude and $\phi$ is latitude.

I wanted to check this myself. I recall (correctly, I hope) that the ratio by which a coordinate transform changes area is given by the magnitude of the determinant of the jacobian matrix of that transformation. For the transform I defined above, this is pretty trivially $\cos{\phi}$, which would imply that the area of a region in projected coordinates would vary depending on it's latitude.

So, I am confused. I clearly misunderstand something or have calculated something incorrectly, because my results would indicate that the sinusoidal projection is trivially not area preserving. What is my error, and how would I have correctly done this calculation?

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You computed the Jacobian determinant correctly. But note that the area element in the $(\lambda,\phi)$ coordinate is not $\mathrm{d}\lambda \mathrm{d}\phi$, but in fact $\cos \phi \mathrm{d}\lambda \mathrm{d}\phi$.

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So in other words, the projection is area preserving precisely because the Jacobian determinant is the same in both coordinates? –  Colin K Apr 15 '11 at 18:47
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Eh... not quite. The point is this: the $(\lambda,\phi)$ (longitude-latitude) coordinate system is curvilinear (as opposed to rectangular). The infinitesimal area carries a factor that comes from "where you are". One slightly bad analogy is the map between "radial coordinates" $(r,\theta)$ and "rectangular coordinates" $(x,y)$ of the plane. The Jacobian determinant between the change of variables is not $1$, but rather $r$. But the "change of coordinates" represent the identiy map from the plane to itself, so area is preserved. –  Willie Wong Apr 15 '11 at 19:02
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What's important is that in geometry, the area is not just given by the coordinate system. You need to associate to the coordinate system also the "area element", which describes how to measure infinitesimally the area with respect to the coordinate system. So the "area preserving" notion is really that "the area described by $(\lambda,\phi)$ with the spherical area element" is identical to "the area described by $(x,y)$ with the rectangular area element". And that statement you check using the Jacobian determinant. –  Willie Wong Apr 15 '11 at 19:06

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