3: Geometries
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- Oct 10, 2021
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- 3.1: Geometries and Models
- An integral part of the modern understanding of geometry is the concept of congruence transformation, or simply symmetry. The symmetries of a geometric space preserve inherent properties of figures, such as distance, angle, and area.
- 3.2: Möbius Geometry
- Möbius geometry provides a unifying framework for studying planar geometries. In particular, the transformation groups of hyperbolic and elliptic geometries in the sections that follow are subgroups of the group of Möbius transformations.
- 3.3: Hyperbolic geometry
- Before the discovery of hyperbolic geometry, it was believed that Euclidean geometry was the only possible geometry of the plane. In fact, hyperbolic geometry arose as a byproduct of efforts to prove that there was no alternative to Euclidean geometry. In this section, we present a Kleinian version of hyperbolic geometry.
- 3.4: Elliptic geometry
- Elliptic geometry is the geometry of the sphere (the 2-dimensional surface of a 3-dimensional solid ball), where congruence transformations are the rotations of the sphere about its center.
- 3.5: Projective Geometry
- Early motivation for the development of projective geometry came from artists trying to solve practical problems in perspective drawing and painting. In this section, we present a modern Kleinian version of projective geometry.
