
32 Chapter 2 The Graphics System
which can be solved to obtain
d = D
.
(X − P)
The point of projection is therefore defined by
Y − P = (D
.
(X − P))D = DD
T
(X − P)
Equivalently, the projection is
Y = DD
T
X +
I − DD
T
P (2.39)
which is of the form Y = MX + B, therefore orthogonal projection onto a line is
an affine transformation. Unlike our previous examples, this transformation is not
invertible. Each point on a line has infinitely many points that project to it (an entire
plane’s worth), so you cannot unproject a point from the line unless you have more
information. Algebraically, the noninvertibility shows up in that M =DD
T
is not an
invertible matrix.
Orthogonal ...