Chain Geometry Determined by the Affine Group

Computational Geometry from the Viewpoint of Affine Differential Geometry

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...

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...

Geometry of Self { Affine Tiles

For a self{similar or self{aane tile in R n we study the following questions: 1) What is the boundary? 2) What is the convex hull? We show that the boundary is a graph directed self{aane fractal, and in the self{similar case we give an algorithm to compute its dimension. We give necessary and suucient conditions for the convex hull to be a polytope, and we give a description of the Gauss map of...

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...