Spread

Spead is a notion from Rational Trigonometry, in many ways related to the idea of angle measurement, without being transcendental or irrational. It is a rational measurement of the separation between vectors. It is defined as

$$ S(v_1,v_2)\equiv1-\frac{(v_1\cdot v_2)^2}{Q(v_1)Q(v_2)} = \frac{(v_1\cdot v_1)(v_2\cdot v_2)-(v_1\cdot v_2)^2}{(v_1\cdot v_1)(v_2\cdot v_2)} $$

In the Euclidean plane, spread goes from 0 to 1, with a maximum when v and w are perpendicular. It is important to note that spread is defined between lines, not rays. For those still clinging to modern trigonometry, it is perhaps helpful to begin by observing that spread is equal to $$\sin^2\theta$$ of an angle.

The spread 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:

$$ c(l_1,l_2)\equiv \frac{(a_1 a_2 - b_1 b_2)^2}{(a_1 ^ 2 + b_1 ^ 2) (a_2 ^ 2 + b_2 ^ 2)} = 1 - \frac{(a_1a_2+b_1b_2)^2}{(a_1^2+b_1^2)(a_2^2+b_2^2)} $$

Spread law
For three points $$ A_1,A_2, $$ and $$ A_3 $$ with non-zero quadrances $$ Q_1 \equiv (A_2,A_3), Q_2 \equiv (A_1,A_3), Q_3 \equiv (A_1,A_2) $$, the spreads $$ s_1 \equiv s(A_1 A_2, A_1 A_3), s_2 \equiv s(A_2 A_1, A_2 A_3), $$ and $$ s_3 \equiv s(A_3 A_1, A_3 A_2) $$ are related by:

$$ \frac{s_1}{Q_1} = \frac{s_2}{Q_2} = \frac{s_3}{Q_3} $$

Uses
Because spread is a squared quantity, it is the same for both parts of a supplementary pair of angles. What if we are on a circle? Dr. Wildberger suggests this is the time to use something more akin to angles, like the tangent half-angle/parameterization of the circle.

$$ \left( \frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2} \right) $$