Linear Algebra, 2020a

Topic: Projective Geometry 385
(We’ve included one of the projective points on this line to bring out a subtlety.
Because two antipodal spots together make up a single projective point, the
great circle’s behind-the-paper part is the same set of projective points as its
in-front-of-the-paper part.) Just as we did with each projective point, we can
also describe a projective line with a triple of reals. For instance, the members
of this plane through the origin in R 3
⎛ ⎞
x
⎜ ⎟
{ ⎝y⎠ | x + y − z = 0}
z
project to a line that we can describe with (1 1 −1) (using a row vector for
this typographically distinguishes lines from points). In general, for any nonzero
three-wide row vector ⃗L we define the associated line in the projective plane,
to be the set L = {k⃗L | k ∈ R and k ≠ 0}.
The reason this description of a line as a triple is convenient is that in
the projective plane a point v andalineL are incident — the point lies on
the line, the line passes through the point — if and only if a dot product of
their representatives v 1 L 1 + v 2 L 2 + v 3 L 3 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 R 3
whose components are in ratio 1 : 2 : 3 lies in the plane through the origin whose
equation is of the form k · x + k · y − k · 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.
With this, we can do analytic projective geometry. For instance, the projective
line L =(1 1 −1) has the equation 1v 1 + 1v 2 − 1v 3 = 0, meaning that for any
projective point v incident with the line, any of v’s representative homogeneous
coordinate vectors will satisfy the equation. This is true simply because those
vectors lie on the three space plane. One difference from Euclidean analytic
geometry is that in projective geometry besides talking about the equation of a
line, we also talk about the equation of a point. For the fixed point
⎛ ⎞
1
⎜ ⎟
v = ⎝2⎠
3
the property that characterizes lines incident on this point is that the components
of any representatives satisfy 1L 1 + 2L 2 + 3L 3 = 0 and so this is the equation of
v.