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.

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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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