Computations with Homogeneous Coordinates - Universal Hyperbolic Geometry

Explore the fundamental concepts of hyperbolic geometry in this 45-minute lecture from the Universal Hyperbolic Geometry series. Delve into the algebraic definitions of points and lines using proportions of three numbers, highlighting the duality between these objects. Learn how to plot points and lines, derive formulas for joining points and finding intersections of lines, and understand the J function for computations. Discover applications to Cartesian geometry and gain insights into the computational nature of hyperbolic geometry. Visualize concepts through three-dimensional representations and grasp the connection between lines and planes in space and their projections on the viewing plane.

CONTENT SUMMARY: Lines and planes through the origin as points and lines on the viewing plane @00:01 A projected line on the viewing plane @05:32 Official definitions: hyperbolic point, hyperbolic line @08:48 examples: plot points, plot lines @14:29 find a line given 2 points @21:55 A graphical illustration: @25:15 page change: solution to prob. on previous page @26:28 Join of two points theorem @28:50 Meet of two lines theorem @31:51 Duality rinciple @34:46 formulas have application to cartesian geometry 37:38 meet of lines app. to cartesian geom. @40:03 hyperbolic Geometry is a computational subject memorize j function @ THANKS to EmptySpaceEnterprise
Introduction
Three dimensional space V³
Definitions projective point and line
Problem 1: Plot points and linesp
Join of two points theorem
Meet of two lines theorem
Duality principle
Application to Cartesian geometry