Скачать или смотреть Computations with homogeneous coordinates | Universal Hyperbolic Geometry 8 | NJ Wildberger

We discuss the two main objects in hyperbolic geometry: points and lines. In this video we give the official definitions of these two concepts: both defined purely algebraically using proportions of three numbers. This brings out the duality between points and lines, and connects with our 3 dimensional picture of lines and planes in the space, or our 3 dimensional picture of the projective plane.

We derive several important theorems: the formulas for the lines joining two points, and dually the point where two lines meet. We introduce the J function for making such computations.

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 @42:18 (THANKS to EmptySpaceEnterprise)

Video Content:
00:00 Introduction
5:23 Three dimensional space V³
8:42 Definitions projective point and line
14:23 Problem 1: Plot points and linesp
28:47 Join of two points theorem
31:47 Meet of two lines theorem
34:40 Duality principle
37:34 Application to Cartesian geometry

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