This section is optional; only the last two sections of Chapter Five require this material.
We have described the projection
π
{\displaystyle \pi }
from
R
3
{\displaystyle \mathbb {R} ^{3}}
into its
x
y
{\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
v
→
{\displaystyle {\vec {v}}}
is the
p
→
{\displaystyle {\vec {p}}}
in the plane with the property that
someone standing on
p
→
{\displaystyle {\vec {p}}}
and looking straight up or down sees
v
→
{\displaystyle {\vec {v}}}
.
In this section we will generalize this to other projections,
both orthogonal (i.e., "straight up and down") and nonorthogonal.