Cross

Cross is the replacement for $$\cos^2\theta$$ in Rational Trigonometry which captures the separation between lines without being transcendental or irrational. The cross between two non-null lines $$l_1 \equiv \langle a_1:b_1:c_1\rangle $$ and $$l_2 \equiv \langle a_2:b_2:c_2 \rangle$$ is the number:

$$ c(l_1,l_2)\equiv \frac{(a_1 a_2 + b_1 b_2)^2}{(a_1 ^ 2 + b_1 ^ 2) (a_2 ^ 2 + b_2 ^ 2)} $$

In the Euclidean plane, cross goes from 0 to 1, with a maximum when $$l_1$$ and $$l_2$$ are parallel.

Spread Plus Cross
The cross is related to the spread by:

$$ c(l_1,l_2) + s(l_1, l_2) = 1 $$

Cross Law
Suppose the three points $$A_1$$, $$A_2$$, and $$A_3$$ form quadrances $$Q_1 \equiv Q(A_2, A_3), Q_2 \equiv Q(A_1, A_3),$$ and $$Q_2 \equiv Q(A_1, A_2)$$ and define the cross $$ c_3 = (A_3 A_1, A_3 A_2) $$ then:

$$(Q_1+Q_2-Q_3)^2=4Q_1Q_2c_3$$