Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

CHAPTER XVIII

THE INFLEXION POINTS OF A PLANE CUBIC CURVE
173. Homogeneous Coordinates of Points in a Plane. Let
ax+b y+c,=0 (i=1, 2, 3)
be any three linear equations such that
al bl cl
A= a2 b2 C2 #0O.
a3 b3 C3
Interpret x and y as the Cartesian coordinates of a point referred
to rectangular axes. Then the three equations represent three
straight lines Li forming a triangle. Choose the sign before
the radical so that
aP x + bey +cy
~ Va/a,2 + b2
is positive for a point (x, y) inside the triangle, and hence is
the length of the perpendicular from that point to Li. The
homogeneous coordinates of a point (x, y) are three numbers
X1, X2, X3 such that
pxl =kip, px2= k2p2, pX3= k3p3,
where kl, k2, k3 are constants, the same for all points, while
p is an arbitrary factor of proportionality. Thus only the
ratios of xi, X2, X3 are defined. The coefficients of the linear
function kip, which are proportional to aj, bi, ci, will henceforth be denoted by those same letters. Then
(1) px =aix+biy+c-, A#O (i=1, 2, 3).
Solving these equations by determinants, we get
Ax=p A jxj, Ay= p Bx A==p Cal,
327