The axioms of descriptive geometry, by A.N. Whitehead.

38-40] INFINITESIMAL PROJECTIVE TRANSFORMATIONS

43

40. Now consider transformations of the type
Tx (allx + a12y + a3z + a14)/(alx + a2,y + a3z + 1)'
Ty = (a2lx +a2y + a 2y z + a a4)/(ax + a2y + az + 1)...(1).
Tz = (as31 + + a 3+ a33 + a34)/(a1x + a2y + a3z + 1)
They obviously belong to the general projective group as defined
above. Also there are fifteen effective parameters. But if we
substitute for x, y, z the coordinates of a given point A, and for
Tx, Ty, Tz the coordinates of a given point A', three equations are
obtained between the parameters. Let the same be done for B and B',
C and C', D and D', E and E'. Then in all fifteen equations are
found. Also if no four of A, B, C, D, E are coplanar, and no four of
A', B', C', D', E' are coplanar, these equations are consistent, and
definitely determine the transformation T. Hence (cf. ~ 39) the
equations (1) can, by a proper choice of parameters, be made to
represent any assigned transformation of the general projective group.
Hence the transformations represented by them are those of the whole
general projective group.
It is obvious from the form of these equations that the group is
a fifteen-limbed continuous transformation-group. To find its infinitesimal transformations, put
al = 1 + allt, 1al2 = at, a13 = a3t, a14 = al4t, a1 =at, a a2t, a1 = at, etc.
Then we find that the analogues of equations (6) of ~ 35 are
dx
d=t alx + a2y + z + a- x (alx + a2y + az)
dy a2lx a22 a, + - + - a y (ax + a2y + z)...... (2).
dt I
dt = a31x + + a az3 + a- z (alx + a2y + a3z)
These equations give the general form of an infinitesimal transformation of the general projective group.