Universal Hyperbolic Geometry

Course Description

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. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry. Look up for his course in order to familiarize with this new development. He also has recorded very organized and detailed lecture series on Algebraic Topology, History of Mathematic, Linear Algebra, and more.

Universal Hyperbolic Geometry
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines. (source: Wikipedia)
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Video Lectures & Study Materials

# Lecture Play Lecture
I. Introduction to Universal Hyperbolic Geometry
1 Introduction to Universal Hyperbolic Geometry (23:13) Play Video
2 Apollonius and Polarity (40:38) Play Video
3 Apollonius and Harmonic Conjugates (38:22) Play Video
4 Pappus' Theorem and the Cross Ratio (21:38) Play Video
5 First Steps in Hyperbolic Geometry (37:14) Play Video
6 The Circle and Cartesian Coordinates (35:54) Play Video
7 Duality, Quadrance and Spread in Cartesian Coordinates (50:38) Play Video
8 The Circle and Projective Homogeneous Coordinates (37:40) Play Video
9 The Circle and Projective Homogeneous Coordinates II (24:17) Play Video
10 Computations with Homogeneous Coordinates (44:32) Play Video
11 Duality and Perpendicularity (33:11) Play Video
12 Existence of Orthocenters (44:06) Play Video
13 Theorems using Perpendicularity (37:28) Play Video
14 Null Points and Null Lines (36:20) Play Video
15 Apollonius and Polarity Revisited (26:31) Play Video
16 Reflections in Hyperbolic Geometry (31:23) Play Video
17 Reflections and Projective Linear Algebra (50:29) Play Video
18 Midpoints and Bisectors (36:41) Play Video
19 Medians, Midlines, Centroids and Circumcenters (34:09) Play Video
20 Parallels and the Double Triangle (29:35) Play Video
21 The J function, sl(2) and the Jacobi identity (42:27) Play Video
22 Pure and Applied Geometry: understanding the continuum (38:44) Play Video
23 Quadrance and Spread (35:54) Play Video
24 Pythagoras' Theorem in Universal Hyperbolic Geometry (36:14) Play Video
25 The Triple Quad Formula in Universal Hyperbolic Geometry (39:11) Play Video
26 Visualizing Quadrance with Circles (34:34) Play Video
27 Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry (25:22) Play Video
28 Trigonometric Laws in Hyperbolic Geometry using Geometer's Sketchpad (20:20) Play Video
29 The Spread Law in Universal Hyperbolic Geometry (24:21) Play Video
30 The Cross Law in Universal Hyperbolic Geometry (35:26) Play Video
31 Thales' Theorem, Right Triangles and Napier's Rules (42:35) Play Video
32 Isosceles Triangles in Hyperbolic Geometry (32:46) Play Video
33 Menelaus, Ceva and the Laws of Proportion (42:05) Play Video
34 Trigonometric Dual Laws and the Parallax Formula (35:47) Play Video
II. Spherical and Elliptic Geometries
35 Introduction to Spherical and Elliptic Geometries (32:14) Play Video
36 Introduction to Spherical and Elliptic Geometries II (44:19) Play Video
37 Areas and Volumes for a Sphere (32:14) Play Video
38 Classical Spherical Trigonometry (34:48) Play Video
39 Perpendicularity, Polarity and Duality on a Sphere (31:54) Play Video
40 Parametrizing and Projecting a Sphere (38:43) Play Video
III. Overview of Rational Trigonometry
41 Rational Trigonometry: An Overview (33:08) Play Video
42 Rational Trigonometry in Three Dimensions (31:20) Play Video

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