School of Mathematics and Statistics, UNSW, Sydney Australia
Abstract:
Rational trigonometry is a framework for Euclidean planar geometry that replaces distance and angle with the quadratic concepts of quadrance and spread, which are algebraically closer to inner products and geometric algebra, and which avoid use of transcendental functions in the framing of the main laws. Many computations that previously resulted in approximate solutions can now be solved with complete accuracy in appropriate field extensions, and can be run in demonstrably shorter times; in principle without calculators, computers or trigonometric tables. This approach allows Euclidean geometry to be extended in a universal way to general fields not of characteristic two.
The theory also generalizes in two other important directions; first of all to general inner product spaces in higher dimensions, even with indefinite inner products, such as the Minkowski space of special relativity; and secondly to a projective setting, where both spherical/elliptic and hyperbolic geometries are subsumed in a uniform way, essentially replacing the formulas of spherical trigonometry with purely algebraic equivalents. This provides a powerful new framework for surveying, crystallography, optics, robotics, visualization and computer graphics. We will illlustrate the practical usefulness of the theory with concrete examples and computations involving the ZOME modelling system in three dimensions.
Have a look on the book:
Wildberger,N.J.: DIVINE
PROPORTIONS: Rational Trigonometry to Universal Geometry