# InfoCoBuild

## Universal Hyperbolic Geometry

Universal Hyperbolic Geometry (UNSW). This is a collection of video lectures on Universal Hyperbolic Geometry given by Professor N. J. Wildberger. This course explains 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.

 Lecture 05 - The Circle and Cartesian Coordinates

This lecture introduces basic facts about points, lines and the unit circle in terms of Cartesian coordinates. We determine the line joining two points, the condition for colinearity of three points (using the determinant), the point where two non-parallel lines meet and the the condition for concurrency of three lines. We state the rational parametrization of the circle and show that a line meets a circle in either 1,2 or 0 points.

Go to the Course Home or watch other lectures:

 Lecture 01 - Apollonius and Polarity Lecture 02 - Apollonius and Harmonic Conjugates Lecture 03 - Pappus' Theorem and the Cross Ratio Lecture 04 - First Steps in Hyperbolic Geometry Lecture 05 - The Circle and Cartesian Coordinates Lecture 06 - Duality, Quadrance and Spread in Cartesian Coordinates Lecture 07 - The Circle and Projective Homogeneous Coordinates Lecture 08 - Computations and Homogeneous Coordinates Lecture 09 - Duality and Perpendicularity Lecture 10 - Orthocenters Exist! Lecture 11 - Theorems Using Perpendicularity Lecture 12 - Null Points and Null Lines Lecture 13 - Apollonius and Polarity Revisited Lecture 14 - Reflections in Hyperbolic Geometry Lecture 15 - Reflections and Projective Linear Algebra Lecture 16 - Midpoints and Bisectors Lecture 17 - Medians, Midlines, Centroids and Circumcenters Lecture 18 - Parallels and the Double Triangle Lecture 19 - The J Function, sl(2) and the Jacobi Identity Lecture 20 - Pure and Applied Geometry - Understanding the Continuum Lecture 21 - Quadrance and Spread Lecture 22 - Pythagoras' Theorem in Universal Hyperbolic Geometry Lecture 23 - The Triple Quad Formula in Universal Hyperbolic Geometry Lecture 24 - Visualizing Quadrance with Circles Lecture 25 - Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry Lecture 26 - Trigonometric Laws in Hyperbolic Geometry using Geometer's Sketchpad Lecture 27 - The Spread Law in Universal Hyperbolic Geometry Lecture 28 - The Cross Law in Universal Hyperbolic Geometry Lecture 29 - Thales' Theorem, Right Triangles and Napier's Rules Lecture 30 - Isosceles Triangles in Hyperbolic Geometry Lecture 31 - Menelaus, Ceva and the Laws of Proportion Lecture 32 - Trigonometric Dual Laws and the Parallax Formula Lecture 33 - Spherical and Elliptic Geometries: An Introduction Lecture 34 - Spherical and elliptic geometries (cont.) Lecture 35 - Areas and Volumes for a Sphere Lecture 36 - Classical Spherical Trigonometry Lecture 37 - Perpendicularity, Polarity and Duality on a Sphere Lecture 38 - Parameterizing and Projecting a Sphere Lecture 39 - Rational Trigonometry: An Overview Lecture 40 - Rational Trigonometry in Three Dimensions