Abstract
Based on the measurement in a Cayley-Klein geometry, we can now define specific geometric objects and relations. For instance a circle may be defined as the set of all points that have a constant distance to a given point. Being orthogonal may be defined as a certain angle relation between two lines. In each type of a Cayley-Klein geometry the objects and relations will have very specific properties. In this chapter we will deal with aspects of elementary geometry in the context of Cayley-Klein geometries. Following the spirit of this book, we will again focus on (algebraic and geometric representations of) geometric primitive operations, on incidence theorems, and on invariance properties. Again we try to present the definitions and statements in a way that they apply as generally as possible to degenerate Cayley Klein geometries. Still some statements may break down if the geometric configurations or the underlying geometry becomes too degenerate. Since we do not want to spend most of the exposition mainly with pathological degenerate cases, we will base our definitions whenever possible on constructive approaches that allow us to explicitly calculate the objects involved.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Richter-Gebert, J. (2011). Cayley-Klein Geometries at Work. In: Perspectives on Projective Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17286-1_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-17286-1_22
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17285-4
Online ISBN: 978-3-642-17286-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)