Affine Geometry and Relativity

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 hyp...

Affine Geometry, Projective Geometry, and Non- Euclidean Geometry

1. Affine Geometry 1.1. Affine Space 1.2. Affine Lines 1.3. Affine transformations 1.4. Affine Collinearity 1.5. Conic Sections 2. Projective Geometry 2.1. Perspective 2.2. Projective Plane 2.3. Projective Transformations 2.4. Projective Collinearity 2.5. Conics 3. Geometries and Groups 3.1. Transformation Groups 3.2. Erlangen Program 4. Non-Euclidean Geometry 4.1. Elliptic Geometry 4.2. Hyperb...

Nonlinear distributional geometry and general relativity

This work reports on the construction of a nonlinear distributional geometry (in the sense of Colombeau’s special setting) and its applications to general relativity with a special focus on the distributional description of impulsive gravitational waves.

Semi-Riemann Geometry and General Relativity

Affine Geometry: a Lattice Characterization

Necessary and sufficient conditions are given for a lattice L to be the lattice of flats of an affine space of arbitrary (possibly infinite) dimension. 1. Incidence spaces and Hilbert lattices. By an incidence space [Gl], we mean a system of points, lines, and planes satisfying Hilbert's Axioms of Incidence [Verknüpfung] [HI] as follows: (11) Any two distinct points determine a unique line. (12...