Apollonius and Polarity - Universal Hyperbolic Geometry

Explore the foundations of Universal Hyperbolic Geometry in this introductory lecture. Delve into Apollonius of Perga's work on conics and circles, examining the crucial concept of polarity. Learn about the polar and pole relationships, the Polar Independence Theorem, and the three-way symmetry in polar configurations. Discover the projective definition of polarity and its application to points inside and outside a circle. Engage with hands-on exercises to reinforce understanding of poles, polars, and quadrangles. Investigate the Polar Duality Theorem and its significance in hyperbolic geometry. Gain insights into this revolutionary approach that extends hyperbolic geometry to arbitrary fields and beyond the light cone.

Introduction
Circles
Polar duality
Polar independence theorem
Proof of theorem
Exercises
Polar duality theorem
Notation