In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form. Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point (α, β) that depend on the imaginary unit i: First system: Second system: Laguerre then interpreted these lines as geodesics: An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero. In other terms, these lines satisfy the differential equation ds2 = 0. On an arbitrary surface one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them isotropic lines. In the complex projective plane, points are represented by homogeneous coordinates and lines by homogeneous coordinates . An isotropic line in the complex projective plane satisfies the equation: In terms of the affine subspace x3 = 1, an isotropic line through the origin is In projective geometry, the isotropic lines are the ones passing through the circular points at infinity. In the real orthogonal geometry of Emil Artin, isotropic lines occur in pairs: A non-singular plane which contains an isotropic vector shall be called a hyperbolic plane. It can always be spanned by a pair N, M of vectors which satisfy We shall call any such ordered pair N, M a hyperbolic pair. If V is a non-singular plane with orthogonal geometry and N ≠ 0 is an isotropic vector of V, then there exists precisely one M in V such that N, M is a hyperbolic pair. The vectors x N and y M are then the only isotropic vectors of V. (Wikipedia).
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From playlist Types of Triangles and Their Properties
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From playlist Types of Triangles and Their Properties
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From playlist Types of Triangles and Their Properties
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How To Construct An Isosceles Triangle
Complete videos list: http://mathispower4u.yolasite.com/ This video will show how to construct an isosceles triangle with a compass and straight edge.
From playlist Triangles and Congruence
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From playlist Properties of Trapezoids
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From playlist Geometry Video Playlist
Pierre Degond: Collective dynamics in life sciences - Lecture 3
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From playlist Mathematical Physics
Santosh Vempala: Reducing Isotropy to KLS: An Almost Cubic Volume Algorithm
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From playlist Workshop: High dimensional measures: geometric and probabilistic aspects
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Mark Hannam (3) - Advanced course in theory and numerics of partial differential equations
PROGRAM: NUMERICAL RELATIVITY DATES: Monday 10 Jun, 2013 - Friday 05 Jul, 2013 VENUE: ICTS-TIFR, IISc Campus, Bangalore DETAL Numerical relativity deals with solving Einstein's field equations using supercomputers. Numerical relativity is an essential tool for the accurate modeling of a wi
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Lecture 1 of Leonard Susskind's Modern Physics concentrating on Cosmology. Recorded January 13, 2009 at Stanford University. This Stanford Continuing Studies course is the fifth of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The t
From playlist Lecture Collection | Modern Physics: Cosmology
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Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar 5/27/22
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Retract rationality of some (exceptional) group varieties by Maneesh Thakur
DATE & TIME 05 November 2016 to 14 November 2016 VENUE Ramanujan Lecture Hall, ICTS Bangalore Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstra
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