Linear Algebra

Topic: Projective Geometry 339
P P P
There are two awkward things about this situation. The first is that neither of
the two points in the domain nearest to the vertical dotted line (see below) has
an image because a projection from those two is along the dotted line that is
parallel to the codomain plane (we sometimes say that these two are projected
“to infinity”). The second awkward thing is that the vanishing point in I isn’t
the image of any point from S because a projection to this point would be along
the dotted line that is parallel to the domain plane (we sometimes say that the
vanishing point is the image of a projection “from infinity”).
For a better model, put the projector P at the origin. Imagine that P is
covered by a glass hemispheric dome. As P looks outward, anything in the line
of vision is projected to the same spot on the dome. This includes things on
the line between P and the dome, as in the case of projection by the movie
projector. It includes things on the line further from P than the dome, as in
the case of projection by the painter. It also includes things on the line that lie
behind P , as in the case of projection by a pinhole.
ℓ = {k ·
� �
1 ��
2 k ∈ R}
3
From this perspective P , all of the spots on the line are seen as the same point.
Accordingly, for any nonzero vector �v ∈ R3 , we define the associated point v
in the projective plane to be the set {k�v � � k ∈ R and k �= 0} of nonzero vectors
lying on the same line through the origin as �v. To describe a projective point
we can give any representative member of the line, so that the projective point
shown above can be represented in any of these three ways.
⎛
⎝ 1
⎞ ⎛
2⎠
⎝
3
1/3
⎞ ⎛
2/3⎠
⎝
1
−2
⎞
−4⎠
−6