= Orthogonal projection area formula
{title2=$dA_{\mathrm{proj}}=|\mathbf n\cdot\mathbf k|\,dA$}
For projection of a planar region with <unit normal> $\mathbf n$ onto a <plane> with <unit normal> $\mathbf k$, the projected <area> is $|\mathbf n\cdot\mathbf k|$ times the original <area>. Indeed, the signed projected <area> of the <parallelogram> spanned by <tangent vectors> $\mathbf u,\mathbf v$ is $(\mathbf u\times\mathbf v)\cdot\mathbf k$, while $\mathbf u\times\mathbf v$ is parallel to $\mathbf n$. Taking magnitudes proves the factor. The same pointwise formula applies to a <parametrized surface>; adding projected patches counts multiplicity unless their images overlap only in sets of zero <area>.
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