Infinity | Projective geometry
In geometry, any hyperplane H of a projective space P may be taken as a hyperplane at infinity. Then the set complement P ∖ H is called an affine space. For instance, if (x1, ..., xn, xn+1) are homogeneous coordinates for n-dimensional projective space, then the equation xn+1 = 0 defines a hyperplane at infinity for the n-dimensional affine space with coordinates (x1, ..., xn). H is also called the ideal hyperplane. Similarly, starting from an affine space A, every class of parallel lines can be associated with a point at infinity. The union over all classes of parallels constitute the points of the hyperplane at infinity. Adjoining the points of this hyperplane (called ideal points) to A converts it into an n-dimensional projective space, such as the real projective space RPn. By adding these ideal points, the entire affine space A is completed to a projective space P, which may be called the projective completion of A. Each affine subspace S of A is completed to a projective subspace of P by adding to S all the ideal points corresponding to the directions of the lines contained in S. The resulting projective subspaces are often called affine subspaces of the projective space P, as opposed to the infinite or ideal subspaces, which are the subspaces of the hyperplane at infinity (however, they are projective spaces, not affine spaces). In the projective space, each projective subspace of dimension k intersects the ideal hyperplane in a projective subspace "at infinity" whose dimension is k − 1. A pair of non-parallel affine hyperplanes intersect at an affine subspace of dimension n − 2, but a parallel pair of affine hyperplanes intersect at a projective subspace of the ideal hyperplane (the intersection lies on the ideal hyperplane). Thus, parallel hyperplanes, which did not meet in the affine space, intersect in the projective completion due to the addition of the hyperplane at infinity. (Wikipedia).
Hyperbola 3D Animation | Objective conic hyperbola | Digital Learning
Hyperbola 3D Animation In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other an
From playlist Maths Topics
11_3_8 Example problem calculating a tangent hyperplane
Let's look at an example where we calculate the function of a tangent hyperplane to a point on a higher dimensional curve.
From playlist Advanced Calculus / Multivariable Calculus
The circle and projective homogeneous coordinates (cont.) | Universal Hyperbolic Geometry 7b
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine
From playlist Universal Hyperbolic Geometry
Computations with homogeneous coordinates | Universal Hyperbolic Geometry 8 | NJ Wildberger
We discuss the two main objects in hyperbolic geometry: points and lines. In this video we give the official definitions of these two concepts: both defined purely algebraically using proportions of three numbers. This brings out the duality between points and lines, and connects with our
From playlist Universal Hyperbolic Geometry
This video provides a description of infinity with several examples. http://mathispower4u.com
From playlist Linear Inequalities in One Variable Solving Linear Inequalities
Definition of infinity In this video, I define the concept of infinity (as used in analysis), and explain what it means for sup(S) to be infinity. In particular, the least upper bound property becomes very elegant to write down. Check out my real numbers playlist: https://www.youtube.co
From playlist Real Numbers
How to visualize infinity in concrete terms.
From playlist Summer of Math Exposition 2 videos
Can You Define the Immeasurable?
What is infinity? Can you define something that, by definition, has no boundaries? A subject extensively studied by philosophers, mathematicians, and more recently, physicists and cosmologists, infinity still stands as an enigma of the intellectual world. We asked people from all walks of
From playlist Mathematics
Singular Hodge Theory for Combinatorial Geometries by Jacob Matherne
PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS: Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE & TIME: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is t
From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)
Daniel Hug: Random tessellations in hyperbolic space - first steps
Random tessellations in Euclidean space are a classical topic and highly relevant for many applications. Poisson hyperplane tessellations present a particular model for which mean values and variances for functionals of interest have been studied successfully and a central limit theory has
From playlist Trimester Seminar Series on the Interplay between High-Dimensional Geometry and Probability
Singular Hodge theory of matroids - Jacob Matherne
Joint IAS/Princeton University Algebraic Geometry Seminar Topic: Singular Hodge theory of matroids Speaker: Jacob Matherne Affiliation: Member, School of Mathematics Date: March 25, 2019 For more video please visit http://video.ias.edu
From playlist Joint IAS/PU Algebraic Geometry
The Time Constant and Limiting Shape in High-Dimensional First Passage Percolation by Si Tang
PROGRAM FIRST-PASSAGE PERCOLATION AND RELATED MODELS (HYBRID) ORGANIZERS Riddhipratim Basu (ICTS-TIFR, India), Jack Hanson (City University of New York, US) and Arjun Krishnan (University of Rochester, US) DATE & TIME 11 July 2022 to 29 July 2022 VENUE Ramanujan Lecture Hall and online Th
From playlist First-Passage Percolation and Related Models 2022 Edited
Lecture 2 | Convex Optimization I (Stanford)
Guest Lecturer Jacob Mattingley covers convex sets and their applications in electrical engineering and beyond for the course, Convex Optimization I (EE 364A). Convex Optimization I concentrates on recognizing and solving convex optimization problems that arise in engineering. Convex se
From playlist Lecture Collection | Convex Optimization
Diophantine Inheritance and dichotomy for P - adic measures by Shreyasi Datta
PROGRAM SMOOTH AND HOMOGENEOUS DYNAMICS ORGANIZERS: Anish Ghosh, Stefano Luzzatto and Marcelo Viana DATE: 23 September 2019 to 04 October 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Ergodic theory has its origins in the the work of L. Boltzmann on the kinetic theory of gases.
From playlist Smooth And Homogeneous Dynamics
http://www.woodthatworks.com/kinetic-sculptures/infinity Infinity Kinetic sculpture by David C. Roy with actual sounds The video segments of the full sculpture have been endited to keep the total length of the video short. For a longer sequence https://youtu.be/nPUcQpyLBh0
From playlist Kinetic Sculpture
Tropical Geometry - Lecture 9 - Tropical Convexity | Bernd Sturmfels
Twelve lectures on Tropical Geometry by Bernd Sturmfels (Max Planck Institute for Mathematics in the Sciences | Leipzig, Germany) We recommend supplementing these lectures by reading the book "Introduction to Tropical Geometry" (Maclagan, Sturmfels - 2015 - American Mathematical Society)
From playlist Twelve Lectures on Tropical Geometry by Bernd Sturmfels
Rolf Schneider: Hyperplane tessellations in Euclidean and spherical spaces
Abstract: Random mosaics generated by stationary Poisson hyperplane processes in Euclidean space are a much studied object of Stochastic Geometry, and their typical cells or zero cells belong to the most prominent models of random polytopes. After a brief review, we turn to analogues in sp
From playlist Probability and Statistics
Orthocenters exist! | Universal Hyperbolic Geometry 10 | NJ Wildberger
In classical hyperbolic geometry, orthocenters of triangles do not in general exist. Here in universal hyperbolic geometry, they do. This is a crucial building block for triangle geometry in this subject. The dual of an orthocenter is called an ortholine---also not seen in classical hyperb
From playlist Universal Hyperbolic Geometry
Minerva Lectures 2012 - Ian Agol Talk 2: The virtual Haken conjecture & geometric group theory
Talk two of the second Minerva lecture series, by Prof. Ian Agol on October 23rd, 2012 at the Mathematics Department, Princeton University. More information available at: http://www.math.princeton.edu/events/seminars/minerva-lectures/minerva-lecture-ii-virtual-haken-conjecture-what-geomet
From playlist Minerva Lectures - Ian Agol