Rational Trigonometry - Quadrance and Spread

Quadrance and Spread

Using quadrance instead of distance and spread instead of angle enables calculations to produce output results whose complexity matches that of the input data. In a typical trigonometry problem, for instance, rational values for quadrances and spreads will lead to calculated results (an unknown spread or quadrance) that will either be rational also or at most an expression containing the roots of only rational numbers. These computational gains (exact results, directly calculable) come at the expense of linearity. Doubling or halving a quadrance or spread does not double or halve as a length or a rotation. Similarly, the sum of two lengths or rotations will not be the sum of their individual quadrances or spreads. The mathematics of Rational trigonometry is, its applications aside, a special instance of a description of geometry in terms of linear algebra (using rational methods such as dot products and quadratic forms.) Students who are first learning trigonometry are often not taught about this use of linear algebra within geometry, and changing this state of affairs is (to paraphrase his comments) one aim of Wildberger's book .

Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with a point defined as an ordered pair (x, y) and a line as a general linear equation