Trigonometric dual laws and the Parallax formula | Universal Hyperbolic Geometry 32 | NJ Wildberger

This video introduces a simple universal analog (called the Right parallax formula) to the Angle of parallelism formula found by N. Lobachevsky and J. Bolyai in classical hyperbolic geometry. First we establish the dual laws of the main trigonometric laws for Universal Hyperbolic Geometry. The Right parallax theorem is proven using the Cross dual law, and we also show how it is related to the classical result of Lobachevsky and Bolyai.

Two further interesting variants are given as Exercises: the Isosceles parallax and General parallax formulas.

Video Content:
00:00 Introduction
6:38 Pythagoras' theorem
10:44 Triple spread formula
13:23 Cross dual law
14:45 Right parallax formula
16:57 Zero quadrance theorem
20:12 Zero spread theorem

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[peterhi]

More original and fascinating results. Thank you as always.

[donitarose]

I feel like I’m on another planet with beautiful math!

[postbodzapism]

Can you give a brief summary on the recent works on UHG that you published over the past 3-4 years? It seemed there are many theorems and I think it would be great if you summarize them in one video, and let viewers complete the picture.

[minh]

Thank you so much for fascinating results. I don't know if we can use this approach on hyperbolic surfaces, for instance double torus, pair of pants, ...