Oriented circles and relativistic geometry II | Geometric Linear Algebra 35 | NJ Wildberger

We continue our discussion of oriented, or signed, or directed circles in the plane, which are also called cycles, and the intimate connection with relativistic geometry in three dimensions. This correspondence makes it easier for us to apply linear algebraic ideas to the geometry of circles, but it also provides an interesting geometrical interpretation of the framework of Einstein's Special Relativity. This theory was developed in the 19th century and is called cyclography, but it is not as well-known these days as it ought to be, and benefits from a purely algebraic development. In this lecture we talk about the homothetic centre of two circles (sometimes also called a center of similitude), how to find this, and then describe a lovely theorem of G. Monge on the homothetic centres of three circles. Tangency of signed circles has a natural interpretation of the relativistic quadrance being zero, and we show how the locus of a circle tangent to two given ones has a relativistic meaning as an intersection of two cones, yielding a hyperbola of centres in the original plane. The famous problem of Apollonius makes its appearance. We describe how spheres in the relativistic space, which appear as hyperboloids of one and two sheets, can be viewed from the point of view of circle geometry, involving interesting pencils of circles. Finally we give a perhaps new theorem that describes the geometrical meaning of the relativistic quadrance between two signed circles. This lecture has a lot of material in it, so go slowly! Video Chapters: 00:00 Introduction 6:22 Lines of circles and homothetic centers 14:05 Monge's theorem 18:21 (Relativistic) quadrance between signed circles 22:15 Quadrance of zero 27:07 Mutually tangent circles 31:42 Problem of Apollonius ******* Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF... Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos)    • Elementary Mathematics Explained (K-6)   (40 videos)    • Ancient Mathematics and insights of Howard...   (7 videos)    • Wild West Banking: A mathematician goes We...   (7 videos)    • Playing Go: the ancient oriental board game   (19 videos)    • Maths and Music   (21 videos)    • Year 9 Mathematics (review fractions, deci...   (10 videos)    • Wild Trig: An introduction to Rational Tri...   (94 videos) Wild Egg Maths Playlists:    • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)