the one-to-one correspondence thus established between cor
responding points of the two ranges is called a projec-
tivity.
To obtain an analytic property of a projectivity, we apply
the sine proportion to two triangles in Fig. 1 and get
AC _sin AVC BC _sin BVC
AV sin ACV’ BV~sin ACV
From these and the formulas with D in place of C, we get
AC AVsinAVC AD_AV sinAVD
BC BV' sin BVC BD~BV sin BVD‘
Hence, by division
AC . AD _ sin A VC # sin A VD
lBC~lBD^sin BVC^sin BVD'
The left member is denoted by {ABCD) and is called the
cross-ratio of the four points taken in this order. Since the
right member depends only on the angles at V, it follows that
{ABCD) = {AiBiCiD l ),
if A h . . . , D i are the intersections of the four rays by a
second transversal. Hence if two ranges are projective, the
cross-ratio of any four points of one range equals the cross
ratio of the corresponding points of the other range.
Let each point of the line AB be determined by its dis
tance and direction from a fixed initial point of the line; let
a be the resulting coordinate of A, and b, c, x those of B,
C, D, respectively. Similarly, let A', B', C', D' have the
coordinates a', b', c', x', referred to a fixed initial point on
their line. Then
Hence
x' — b' _ u x — b j _c
» # i fc
■a c —a
x —a
x — a
V, not on their
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