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Hyperbolic geometry is a type of non-Euclidean geometry that arose historically when mathematicians tried to simplify the axioms of Euclidean geometry, and instead discovered unexpectedly that changing one of the axioms to its negation actually produced a consistent theory. The [course_title] provides an introduction to hyperbolic geometry. It starts by discussing what is meant by “distance” and what is “straight” about a straight line in the Euclidean plane R2. Then it gives an introduction to the hyperbolic plane. Topics include distance and area in the hyperbolic plane, distance-preserving maps, hyperbolic trigonometry and hyperbolic polygons.

### Assessment

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

### Certification

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Course Credit: Open Culture

### Course Curriculum

 Module: 01 Universal Hyperbolic Geometry 0: Introduction 00:23:00 UnivHypGeom1: Apollonius and polarity 00:40:00 UnivHypGeom2: Apollonius and harmonic conjugates 00:38:00 UnivHypGeom3: Pappus’ theorem and the cross ratio 00:21:00 UnivHypGeom4: First steps in hyperbolic geometry 00:37:00 UnivHypGeom5: The circle and Cartesian coordinates 00:36:00 Module: 02 UnivHypGeom6: Duality, quadrance and spread in Cartesian coordinates 00:51:00 UnivHypGeom7a: The circle and projective homogeneous coordinates 00:37:00 UnivHypGeom7b: The circle and projective homogeneous coordinates (cont.) 00:24:00 UnivHypGeom8: Computations with homogeneous coordinates 00:44:00 UnivHypGeom9: Duality and perpendicularity 00:33:00 UnivHypGeom10: Orthocenters exist! 00:44:00 Module: 03 UnivHypGeom11: Theorems using perpendicularity 00:37:00 UnivHypGeom12: Null points and null lines 00:36:00 UnivHypGeom13: Apollonius and polarity revisited 00:26:00 UnivHypGeom14: Reflections in hyperbolic geometry 00:31:00 UnivHypGeom15: Reflections and projective linear algebra 00:50:00 UnivHypGeom16: Midpoints and bisectors 00:37:00 UnivHypGeom17: Medians, midlines, centroids and circumcenters 00:34:00 Module: 04 UnivHypGeom18: Parallels and the double triangle 00:29:00 UnivHypGeom19: The J function, sl(2) and the Jacobi identity 00:42:00 UnivHypGeom20: Pure and applied geometry–understanding the continuum 00:39:00 UnivHypGeom21: Quadrance and spread 00:36:00 UnivHypGeom22: Pythagoras’ theorem in Universal Hyperbolic Geometry 00:36:00 UnivHypGeom23: The Triple quad formula in Universal Hyperbolic Geometry 00:39:00 Module: 05 UnivHypGeom24: Visualizing quadrance with circles 00:34:00 UnivHypGeom25: Geometer’s Sketchpad and circles in Universal Hyperbolic Geometry 00:25:00 UnivHypGeom26: Trigonometric laws in hyperbolic geometry using Geometer’s Sketchpad 00:20:00 UnivHypGeom27: The Spread law in Universal Hyperbolic Geometry 00:24:00 UnivHypGeom28: The Cross law in Universal Hyperbolic Geometry 00:35:00 UnivHypGeom29: Thales’ theorem, right triangles and Napier’s rules 00:42:00 Module: 06 UnivHypGeom30: Isosceles triangles in hyperbolic geometry 00:33:00 UnivHypGeom31: Menelaus, Ceva and the Laws of proportion 00:42:00 UnivHypGeom32: Trigonometric dual laws and the Parallax formula 00:36:00 UnivHypGeom33: Spherical and elliptic geometries: an introduction 00:32:00 UnivHypGeom34: Spherical and elliptic geometries (cont.) 00:44:00 UnivHypGeom35: Areas and volumes for a sphere 00:32:00 Module: 07 UnivHypGeom36: Classical spherical trigonometry 00:35:00 UnivHypGeom37: Perpendicularity, polarity and duality on a sphere 00:32:00 UnivHypGeom38: Parametrizing and projecting a sphere 00:39:00 UnivHypGeom39: Rational trigonometry: an overview 00:33:00 UnivHypGeom40: Rational trigonometry in three dimensions 00:31:00 Assessment Submit Your Assignment 00:00:00 Certification 00:00:00

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