Pappus' Theorem and Cross Ratio in Projective Geometry - Universal Hyperbolic Geometry 3

Explore the fundamental concepts of projective geometry in this 22-minute video lecture from the Universal Hyperbolic Geometry series. Delve into Pappus' theorem, the cornerstone of projective geometry, and understand the crucial concept of cross ratio for points on a line and concurrent lines. Learn about the Cross ratio theorem, which demonstrates the invariance of cross ratio under projection, and Chasles theorem for points on a conic. Examine the applications of these principles in hyperbolic geometry and gain insights into the work of Pappus of Alexandria. Practice with exercises to verify Pappus' theorem in special cases and develop a deeper understanding of cross ratio transformations. Conclude by exploring the connections between projective geometry and Desargues' theorem, providing a comprehensive overview of these fundamental geometric concepts.

CONTENT SUMMARY: Pappus' theorem @00:52 cross ratio @02:46 cross ratio transformation theorem @11:08 cross ratio theorem @13:54 Chasles theorem @ The cross ratio is the most important invariant in projective geometry 9:09
Hyperbolic geometry as based on projective geometry
Pappus of Alexandria
Exercise - Verify Pappus' theorem for special cases
Cross ratio
Experience with cross ratio
Cross ratio transformation theorem
Cross ratio theorem
Chasles' theorem
Projective geometry and Desargues