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.