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An Intuitive Introduction to Projective Geometry Using Linear Algebra ΡΠΊΠ°Ρ‡Π°Ρ‚ΡŒ Π² Ρ…ΠΎΡ€ΠΎΡˆΠ΅ΠΌ качСствС

An Intuitive Introduction to Projective Geometry Using Linear Algebra 3 Π³ΠΎΠ΄Π° Π½Π°Π·Π°Π΄

ΡΠΊΠ°Ρ‡Π°Ρ‚ΡŒ Π²ΠΈΠ΄Π΅ΠΎ

ΡΠΊΠ°Ρ‡Π°Ρ‚ΡŒ mp3

ΡΠΊΠ°Ρ‡Π°Ρ‚ΡŒ mp4

ΠΏΠΎΠ΄Π΅Π»ΠΈΡ‚ΡŒΡΡ

Ρ‚Π΅Π»Π΅Ρ„ΠΎΠ½ с ΠΊΠ°ΠΌΠ΅Ρ€ΠΎΠΉ

Ρ‚Π΅Π»Π΅Ρ„ΠΎΠ½ с Π²ΠΈΠ΄Π΅ΠΎ

бСсплатно

Π·Π°Π³Ρ€ΡƒΠ·ΠΈΡ‚ΡŒ,

НС удаСтся Π·Π°Π³Ρ€ΡƒΠ·ΠΈΡ‚ΡŒ Youtube-ΠΏΠ»Π΅Π΅Ρ€. ΠŸΡ€ΠΎΠ²Π΅Ρ€ΡŒΡ‚Π΅ Π±Π»ΠΎΠΊΠΈΡ€ΠΎΠ²ΠΊΡƒ Youtube Π² вашСй сСти.
ΠŸΠΎΠ²Ρ‚ΠΎΡ€ΡΠ΅ΠΌ ΠΏΠΎΠΏΡ‹Ρ‚ΠΊΡƒ...
An Intuitive Introduction to Projective Geometry Using Linear Algebra
  • ΠŸΠΎΠ΄Π΅Π»ΠΈΡ‚ΡŒΡΡ Π’Πš
  • ΠŸΠΎΠ΄Π΅Π»ΠΈΡ‚ΡŒΡΡ Π² ОК
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Π‘ΠΊΠ°Ρ‡Π°Ρ‚ΡŒ Π²ΠΈΠ΄Π΅ΠΎ с ΡŽΡ‚ΡƒΠ± ΠΏΠΎ ссылкС ΠΈΠ»ΠΈ ΡΠΌΠΎΡ‚Ρ€Π΅Ρ‚ΡŒ Π±Π΅Π· Π±Π»ΠΎΠΊΠΈΡ€ΠΎΠ²ΠΎΠΊ Π½Π° сайтС: An Intuitive Introduction to Projective Geometry Using Linear Algebra Π² качСствС 4k

Π£ нас Π²Ρ‹ ΠΌΠΎΠΆΠ΅Ρ‚Π΅ ΠΏΠΎΡΠΌΠΎΡ‚Ρ€Π΅Ρ‚ΡŒ бСсплатно An Intuitive Introduction to Projective Geometry Using Linear Algebra ΠΈΠ»ΠΈ ΡΠΊΠ°Ρ‡Π°Ρ‚ΡŒ Π² максимальном доступном качСствС, Π²ΠΈΠ΄Π΅ΠΎ ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠ΅ Π±Ρ‹Π»ΠΎ Π·Π°Π³Ρ€ΡƒΠΆΠ΅Π½ΠΎ Π½Π° ΡŽΡ‚ΡƒΠ±. Для Π·Π°Π³Ρ€ΡƒΠ·ΠΊΠΈ Π²Ρ‹Π±Π΅Ρ€ΠΈΡ‚Π΅ Π²Π°Ρ€ΠΈΠ°Π½Ρ‚ ΠΈΠ· Ρ„ΠΎΡ€ΠΌΡ‹ Π½ΠΈΠΆΠ΅:

  • Π˜Π½Ρ„ΠΎΡ€ΠΌΠ°Ρ†ΠΈΡ ΠΏΠΎ Π·Π°Π³Ρ€ΡƒΠ·ΠΊΠ΅:

Π‘ΠΊΠ°Ρ‡Π°Ρ‚ΡŒ mp3 с ΡŽΡ‚ΡƒΠ±Π° ΠΎΡ‚Π΄Π΅Π»ΡŒΠ½Ρ‹ΠΌ Ρ„Π°ΠΉΠ»ΠΎΠΌ. БСсплатный Ρ€ΠΈΠ½Π³Ρ‚ΠΎΠ½ An Intuitive Introduction to Projective Geometry Using Linear Algebra Π² Ρ„ΠΎΡ€ΠΌΠ°Ρ‚Π΅ MP3:


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An Intuitive Introduction to Projective Geometry Using Linear Algebra

This is an area of math that I've wanted to talk about for a long time, especially since I have found how projective geometry can be used to formulate Euclidean, spherical, and hyperbolic geometries, and a possible (and hopefully plausible) way projective geometry (specifically the model that uses lines, planes, etc. through the origin) could have been discovered and not just created out of thin air. I am most likely not the first person to discover what I say in this video, but I have not found any sources that explicitly state the same things (except possibly NJ Wildberger with his video on how hyperbolic geometry is "projective relativistic geometry", which I haven't watched, but judging from the thumbnail it seems like he found the same connection between projective geometry and the Minkowski model of hyperbolic geometry that I make in this video). The first half of this video is intended for everyone; the second half (where I start talking about linear algebra) is intended for those who already know that subject on an introductory level, e.g. those who have taken a class in it or have watched 3Blue1Brown's series on it. Everything in this video comes from bits and pieces of articles and videos that I have sporadically watched over the last several (maybe 6 or 7) years, plus linear algebra that I have learned in a class I took more recently. As a result, I probably cannot give a complete list of all the sources I have used, but I will list as many as I can remember down below: Projective geometry: https://en.wikipedia.org/wiki/Homogen... Β Β Β β€’Β ProjectiveΒ geometryΒ andΒ homogeneousΒ coordi...Β Β  ("Projective geometry and homogeneous coordinates | WildTrig: Intro to Rational Trigonometry", Insights into Mathematics) Spherical geometry: https://en.wikipedia.org/wiki/Spheric... https://brilliant.org/wiki/spherical-... Hyperbolic geometry: https://en.wikipedia.org/wiki/Hyperbo... Β Β Β β€’Β ProjectionΒ fromΒ HyperboloidΒ toΒ theΒ Beltram...Β Β  ("Projection from Hyperboloid to the Beltrami–Klein disk.", Jamnitzer) https://dl.tufts.edu/concern/pdfs/bk1... ("Hyperbolic Geometry on a Hyperboloid", William F. Reynolds) https://www.roguetemple.com/z/hyper/m... ("Models and projections of hyperbolic geometry", Rogue Temple) 2D and 3D plots were made with Desmos and GeoGebra, respectively. All other images were made by me in Google Slides. Chapters: PART 1 0:00 Intro 0:31 Defining projective points and lines 4:19 Spatial coordinates 7:11 Projective quadratics 8:40 Non-Euclidean geometries 10:52 Distance metrics 12:11 PART 2 (linear algebra) 12:33 Defining projective points, lines with linear algebra 13:47 clmspace vs. nullspace representation of projective linear objects (points, lines, planes, ...) 16:32 clmspace to nullspace representation of a projective line (includes cross product) 20:31 Spans of clmspaces and interseections of nullspaces 21:33 3D projective geometry 23:13 Projective quadratics and double-cones 26:34 Summary #SoME2

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