Introduction via Flat Affine Planes
The geometries we shall be dealing with in this second part of the book are perhaps the nicest possible infinite geometries. In general, their point sets are well-known two-dimensional surfaces like the Euclidean plane, the real projective plane, the sphe
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The geometries we shall be dealing with in this second part of the book are perhaps the nicest possible infinite geometries. In general, their point sets are well-known two-dimensional surfaces like the Euclidean plane, the real projective plane, the sphere, the torus, and the cylinder. Their lines, blocks, or circles are curves that are nicely embedded in these surfaces. Usually these curves will be homeomorphic to the real line or to the unit circle. The Euclidean plane, viewed as a geometry, and the geometry of circles on the unit sphere are prime examples of the kinds of geometries we will be looking at. The standard references for geometries on surfaces are [42], [89], [90], and [103].
16.1
Some More Basic Facts and Conventions
A topological line on a surface homeomorphic to the xy-plane is a Jordan curve homeomorphic to an open interval that separates the plane into two open component just as a Euclidean line does. A topological circle on a surface is a simply closed Jordan curve on the surface. The Euclidean circles on the sphere are examples of topological circles. A topological line on a surface not homeomorphic to the xy-plane is a topological circle on the surface minus one of its points. A topological sphere and a topological disk are topological spaces homeomorphic to the unit sphere and the interior of the unit circle, respectively. B. Polster, A Geometrical Picture Book © Springer Science+Business Media New York 1998
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16. Introduction via Flat Affine Planes
16.2 The Euclidean Plane-A Flat Affine Plane For most of us, pictures are pictures in the Euclidean plane, and letters and sentences are letters and sentences printed on a small patch of the Euclidean plane. Our pictures of finite geometries in the first part of this book are just special kinds of representations in this prototype geometry that everybody is so familiar with. Anyway, for us the Euclidean plane is a geometry that is a so-called R 2 -plane, that is, a geometry whose point set is homeomorphic to the xy-plane and everyone of whose lines is a topological line in the point set. Furthermore, it satisfies the Axiom (of joining) AI; that is, two points in the point set are contained in exactly one line (see Section 1.3 on affine planes). In fact, the Euclidean plane is a fiat affine plane, that is, an R2-plane that also satisfies Axiom A2 (the "parallel axiom"). Flat affine planes also automatically satisfy Axiom A3 and are therefore affine planes, as we defined them in Section 1.3.
16.2.1
Proper R 2 -Planes
It is easy to construct R 2-planes that are not affine planes. Consider, for example, the restriction of the Euclidean plane to an open vertical strip or, more generally, to an open strictly convex region. Then we are still dealing with an R2-plane. Nevertheless, unless this region is the whole plane, the parallel axiom will never be satisfied. In the following two examples the two thin lines intersect and are both parallel to the thick line. This never happens in affine planes.
16.2.2 Pencils, Parallel Classes, Generator-
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