This section is optional; only the last two sections of Chapter Five require this material.

We have described the projection ${\displaystyle \pi }$ from ${\displaystyle \mathbb {R} ^{3}}$ into its ${\displaystyle xy}$ plane subspace as a "shadow map". This shows why, but it also shows that some shadows fall upward.

So perhaps a better description is: the projection of ${\displaystyle {\vec {v}}}$ is the ${\displaystyle {\vec {p}}}$ in the plane with the property that someone standing on ${\displaystyle {\vec {p}}}$ and looking straight up or down sees ${\displaystyle {\vec {v}}}$. In this section we will generalize this to other projections, both orthogonal (i.e., "straight up and down") and nonorthogonal.