Linear Algebra

Topic: Projective Geometry 341
of this plane through the origin in R3 ⎛
{ ⎝ x
⎞
y⎠
z
� � x + y − z =0}
project to a line that we can described with the triple � 1 1 −1 � (we use row
vectors to typographically distinguish lines from points). In general, for any
nonzero three-wide row vector � L we define the associated line in the projective
plane, tobethesetL = {k� L � � k ∈ R and k �= 0} of nonzero multiples of L. �
The reason that this description of a line as a triple is convienent is that
in the projective plane, a point v and a line L are incident — the point lies
on the line, the line passes throught the point — if and only if a dot product
of their representatives v1L1 + v2L2 + v3L3 is zero (Exercise 4 shows that this
is independent of the choice of representatives �v and � L). For instance, the
projective point described above by the column vector with components 1, 2,
and 3 lies in the projective line described by � 1 1 −1 � , simply because any
vector in R3 whose components are in ratio 1 : 2 : 3 lies in the plane through the
origin whose equation is of the form 1k · x +1k · y − 1k · z = 0 for any nonzero k.
That is, the incidence formula is inherited from the three-space lines and planes
of which v and L are projections.
Thus, we can do analytic projective geometry. For instance, the projective
line L = � 1 1 −1 � has the equation 1v1 +1v2 − 1v3 = 0, because points
incident on the line are characterized by having the property that their representatives
satisfy this equation. One difference from familiar Euclidean anlaytic
geometry is that in projective geometry we talk about the equation of a point.
For a fixed point like
⎛ ⎞
1
v = ⎝2⎠
3
the property that characterizes lines through this point (that is, lines incident
on this point) is that the components of any representatives satisfy 1L1 +2L2 +
3L3 = 0 and so this is the equation of v.
This symmetry of the statements about lines and points brings up the Duality
Principle of projective geometry: in any true statement, interchanging ‘point’
with ‘line’ results in another true statement. For example, just as two distinct
points determine one and only one line, in the projective plane, two distinct
lines determine one and only one point. Here is a picture showing two lines that
cross in antipodal spots and thus cross at one projective point.
Contrast this with Euclidean geometry, where two distinct lines may have a
unique intersection or may be parallel. In this way, projective geometry is
simpler, more uniform, than Euclidean geometry.
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