Affine and Projective Universal Geometry

By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are introduced here, and the main formulas extend those of rational trigonometry in the plane. This gives a unified, computational model of both spherical and hyperbolic geometries, allows the extension of many results of Euclidean geometry to the relativistic setting, and provides a new metrical approach to algebraic geometry.