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source: UNSWelearning 2011年5月1日

Hyperbolic Geometry - N J Wildberger

This is a complete and relatively elementary course explaining a new, simpler and more elegant theory of non-Euclidean geometry; in particular hyperbolic geometry. It is a purely algebraic approach which avoids transcendental functions like log, sin, tanh etc, relying instead on high school algebra and quadratic equations. The theory is more general, extending beyond the null circle, and connects naturally to Einstein's special theory of relativity.

This course is meant for mathematics majors, bright high school students, high school teachers, engineers, scientists, and others with an interest in mathematics and some basic algebraic skills.

Norman Wildberger is also the discoverer of Rational Trigonometry, an important new direction for classical trigonometry (and which really ought to be revolutionizing mathematics education!!)

His YouTube site Insights into Mathematics at user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.

Universal Hyperbolic Geometry 0: Introduction 23:13 This is the introductory lecture to a series on hyperbolic geometry which introduces a radically new and improved way of treating the subject, making it more algebraic and logical, with improved computational power and many new theorems. In this lecture we summarize the differences between this UNIVERSAL HYPERBOLIC GEOMETRY and traditional courses taught at universities. We briefly review some of the framework introduced by Lobachevsky, Bolyai and Gauss.

1: Apollonius and polarity 40:38

2: Apollonius and harmonic conjugates 38:22

3: Pappus' theorem and the cross ratio 21:38

4: First steps in hyperbolic geometry 37:14

5: The circle and Cartesian coordinates 35:54

6: Duality, quadrance and spread in Cartesian coordinates 50:38

7a: The circle and projective homogeneous coordinates 37:40

7b: The circle and projective homogeneous coordinates (cont.) 24:17

8: Computations with homogeneous coordinates 44:32

9: Duality and perpendicularity 33:11

10: Orthocenters exist! 44:06

11: Theorems using perpendicularity 37:28

12: Null points and null lines 36:20

13: Apollonius and polarity revisited 26:31

14: Reflections in hyperbolic geometry 31:23

15: Reflections and projective linear algebra 50:29

16: Midpoints and bisectors 36:41

17: Medians, midlines, centroids and circumcenters 34:09

18: Parallels and the double triangle 29:35

19: The J function, sl(2) and the Jacobi identity 42:27

20: Pure and applied geometry--understanding the continuum 38:44

21: Quadrance and spread 35:54

22: Pythagoras' theorem in Universal Hyperbolic Geometry 36:14

23: The Triple quad formula in Universal Hyperbolic Geometry 39:11

24: Visualizing quadrance with circles 34:34

25: Geometer's Sketchpad and circles in Universal Hyperbolic Geometry 25:22

26: Trigonometric laws in hyperbolic geometry using Geometer's Sketchpad 20:20

27: The Spread law in Universal Hyperbolic Geometry 24:21

28: The Cross law in Universal Hyperbolic Geometry 35:26

29: Thales' theorem, right triangles and Napier's rules 42:35

30: Isosceles triangles in hyperbolic geometry 32:46

31: Menelaus, Ceva and the Laws of proportion 42:05

32: Trigonometric dual laws and the Parallax formula 35:47

33: Spherical and elliptic geometries: an introduction 32:14

34: Spherical and elliptic geometries (cont.) 44:19

35: Areas and volumes for a sphere 32:14

36: Classical spherical trigonometry 34:48

37: Perpendicularity, polarity and duality on a sphere 31:54

38: Parametrizing and projecting a sphere 38:43

39: Rational trigonometry: an overview 33:08

40: Rational trigonometry in three dimensions 31:20

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