370 12. MODERN GEOMETRIESto choose ì so that there will be a seventh common zero. With that choice of ì,
the polynomial t(x) must have seven zeros, and hence must vanish identically. But
since t(x) was the result of eliminating y between the two equations q(x, y) — 0
and s(x, y) — 0, it now follows that q(x, y) divides s(x, y) (see Problem 12.5 below).
That is, the equation s(x, y) = 0 can be written as (ax + by + c)q(x, y) — 0. Hence
its solution set consists of the conic and the line ax + by + c = 0, and this line must
contain the other three points of intersection.
Conic sections and quadratic functions in general continued to be a source of
new ideas for geometers during the early nineteenth century. Plucker liked to use
homogeneous coordinates to give a symmetric description of a quadric surface. To
take the simplest example, consider the sphere of radius 2 in three-dimensional
space with center at (2,3,1), whose equation is
(x-2)^2 + (2/-3)^2 + (2-l)^2 = 4.If x, y, and z, are replaced by î/ô, ç/ô, and æ/ô and each term is multiplied by
r^2 , this equation becomes a homogeneous quadratic relation in the four variables
(i,»?,C,r):
(î - 2r)^2 + (17- 3r)^2 + (C - ô)^2 = 4r^2.
The sphere of unit radius centered at the origin then has the simple equation
ô^2 — î^2 — ç^2 — æ^2 = 0. Plucker introduced homogeneous coordinates in 1830. One
of their advantages is that if r = 0, but the other three coordinates are not all zero,
the point (î, ç, æ, ô) can be considered to be located on a sphere of infinite radius.
The point (0,0,0,0) is excluded, since it seems to correspond to all points at once.
Homogeneous coordinates correspond very well to the ideas of projective geom-
etry, in which a point in a plane is identified with all the points in three-dimensional
space that project to that point from a point outside the plane. If, for example, we
take the center of projection as (0,0,0) and identify the plane with the plane 2 = 1,
that is, each point (x, y) is identified with the point (x, y, 1), the points that project
to (x,y) are all points (tx,ty, t), where t ö 0. Since the equation of a line in the
(x, y)-plane has the form ax+by+c = 0, one can think of the coordinates (a, b, c) as
the coordinates of the line. Here again, multiplication by a nonzero constant does
not affect the equation, so that these coordinates can be identified with (ta, tb, tc)
for any t ö 0. Notice that the condition for the point (x, y) to lie on the line (o, 6, c)
is that ((a, b, c), (x, y, 1)) = a · ÷ + b • y + c • 1 = 0, and this condition is unaffected
by multiplication by a constant. The duality between points and lines in a plane
is then clear. Any triple of numbers, not all zero, can represent either a point or a
line, and the incidence relation between a point and a line is symmetric in the two.
We might as well say that the line lies on the point as that the point lies on the
line.
Equations can be written in either line coordinates or point coordinates. For
example, the equation of an ellipse can be written in homogeneous point coordinates
(£,V, æ) as
bV^ + aVr?^2 = á^2 ß>^2 æ^2 ,
or in line coordinates (ë, ì, í) as
2\2 , r2 2 2 2
oA + b ì = c í ,
where the geometric meaning of this last expression is that the line (ë, ì, u) is
tangent to the ellipse.