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§1. Rational Lines in the Projective Plane 3

Instead of using parallel lines to relate points on two lines L = C f and L = C f ,

we can use a point P 0 = (x 0 , y 0 ) not on either L or L and relate points using the

family of all lines through P 0 . The pair P on L and P on L correspond when P,
P ,and P 0 are all on a line.

If L and L are rational lines, and if P 0 is a rational point, then for two corresponding

points P on L and P on L the point P is rational if and only if P is rational, and

this defines a bijection between C f (Q) and C f (Q).

Observe that there are special cases of lines through P 0 , i.e., those parallel to L

or L , which as matters stand do not give a corresponding pair of points between L

and L . This is related to the fact that the two types of correspondence with parallel
lines and with lines through a point are really the same when viewed in terms of the
projective plane, for parallel lines intersect at a point on the “line at infinity.” As we
see in the next paragraphs, the projective plane is the ordinary Cartesian or affine
plane together with an additional line called the line at infinity.
(1.2) Definition. The projective plane P 2 is the set of all triples w : x : y, where
w, x,and y are not all zero and the points w : x : y and w : x : y are considered

equal provided there is a nonzero constant k with

w = kw, x = kx, y = ky.
As with the affine plane and plane curves we have three basic cases
P 2 (Q) ⊂ P 2 (R) ⊂ P 2 (C)
consisting of triples proportional to w : x : y, where w, x, y ∈ Q for P 2 (Q), where
w, x, y ∈ R for P 2 (R), and where w, x, y ∈ C for P 2 (C).

Note w : x : y ∈ P 2 (C) is also in P 2 (Q) if and only if w, x, y ∈ C can be
rescaled to be elements of Q.

(1.3) Remarks. A line C f in P 2 is the locus of all w : x : y satisfying the equation
F(w, x, y) = aw + bx + cy = 0. The line at infinity L ∞ is given by the equation
w = 0. A point in P 2 − L ∞ has the form 1 : x : y after multiplying with the
factor w −1 . The point 1 : x : y in the projective plane corresponds to (x, y) in the

usual Cartesian plane. For a line L given by aw + bx + cy = 0and L given by

a w + b x + c y = 0wehave L = L if and only if a : b : c = a : b : c in the