Quaternions and spatial rotation

Quaternions EXPLAINED Briefly

This is a video I have been wanting to make for some time, in which I discuss what the quaternions are, as mathematical objects, and how we do calculations with them. In particular, we will see how the fundamental equation of the quaternions i^2=j^2=k^2=ijk=-1 easily generates the rule for

From playlist Quaternions

Quaternions as 4x4 Matrices - Connections to Linear Algebra

In math, it's usually possible to view an object or concept from many different (but equivalent) angles. In this video, we will see that the quaternions may be viewed as 4x4 real-valued matrices of a special form. What is interesting here is that if you know how to multiply matrices, you a

From playlist Quaternions

3D Rotations and Quaternion Exponentials: Special Case

In this video, we'll understand 3D rotations from the point of view of vector analysis and quaternions. We will solve the problem of rotating a vector which is perpendicular to the axis of rotation in this video which will help us solve the general case in the next video. We will especiall

From playlist Quaternions

3D Rotations in General: Rodrigues Rotation Formula and Quaternion Exponentials

In this video, we will discover how to rotate any vector through any axis by breaking up a vector into a parallel part and a perpendicular part. Then, we will use vector analysis (cross products and dot products) to derive the Rodrigues rotation formula and finish with a quaternion point o

From playlist Quaternions

The rotation problem and Hamilton's discovery of quaternions IV | Famous Math Problems 13d

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connec

From playlist Famous Math Problems

Geometric Algebra - Rotors and Quaternions

In this video, we will take note of the even subalgebra of G(3), see that it is isomorphic to the quaternions and, in particular, the set of rotors, themselves in the even subalgebra, correspond to the set of unit quaternions. This brings the entire subject of quaternions under the heading

From playlist Math

The rotation problem and Hamilton's discovery of quaternions III | Famous Math Problems 13c

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this rem

From playlist Famous Math Problems

3D Reflections with Vectors and Quaternions

In addition to rotating vectors, there is a way to reflect vectors through planes using quaternions. We'll derive how to do this transformation from the point of view of standard vector analysis and then from the point of view of quaternions. The quaternion formula is actually quite compac

From playlist Quaternions

Could These Numbers Unravel New Dimensions in Space?

These multidimensional number systems are helping us explain the laws of nature. Here’s how. Can Hawking’s Black Hole Paradox Be Solved With Fuzzballs? - https://youtu.be/esPa1tVSjew Read More: The Peculiar Math That Could Underlie The Laws of Nature https://www.quantamagazine.org/the-

From playlist Elements | Season 4 | Seeker

Lecture 06: 3D Rotations and Complex Representations (CMU 15-462/662)

Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz2emSh0UQ5iOdT2xRHFHL7E Course information: http://15462.courses.cs.cmu.edu/

From playlist Computer Graphics (CMU 15-462/662)

From Hamilton’s Quaternions to Graves & Cayley’s Octonions – Louis Kauffman

We describe geometric and topological approaches to Hamilton's Quaternions and to the Octonions of Graves and Cayley.

From playlist G4G12 Videos

Example of Quaternions

Matrix Theory: For the quaternion alpha = 1 - i + j - k, find the norm N(alpha) and alpha^{-1}. Then write alpha as a product of a length and a direction.

From playlist Matrix Theory

Matching Paired Sets of Space and Orientation Data

To learn more about Wolfram Technology Conference, please visit: https://www.wolfram.com/events/technology-conference/ Speaker: Andrew Hanson Wolfram developers and colleagues discussed the latest in innovative technologies for cloud computing, interactive deployment, mobile devices, and

From playlist Wolfram Technology Conference 2018

Visualizing quaternions (4d numbers) with stereographic projection

How to think about this 4d number system in our 3d space. Part 2: https://youtu.be/zjMuIxRvygQ Interactive version of these visuals: https://eater.net/quaternions Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of t

From playlist Explainers

Feb. 25, Chapter 40 (Minkowski space and the Lorentz group)

From playlist Spring 2021 Course

QED Prerequisites Geometric Algebra 26 The Lorentz Group

In this lesson we connect the idea of using rotors to transform multivectors with the notion of the Lorentz group. Please consider supporting this channel on Patreon: https://www.patreon.com/XYLYXYLYX The software I usually use to produce the lectures is: https://apps.apple.com/us/app

From playlist QED- Prerequisite Topics

GAME2020 - 1. Dr. Leo Dorst. Get Real! (new audio!)

Dr. Leo Dorst from the University of Amsterdam explains how Geometric Algebra subsumes/extends/invigorates Linear Algebra. More information at https://bivector.net This version has an updated audio track.

From playlist Bivector.net

A Swift Introduction to Geometric Algebra

This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realiz

From playlist Miscellaneous Math

quaternion square root of -1

quaternion square root of -1. We calculate the square root of -1 using the quaternions, which involves knowing how to multiply quaternion numbers. The answer will surprise you, because it involves spheres and it will make you see complex numbers in a new way, as north and south poles of ba

From playlist Complex Analysis

AI Weekly Update - April 12th, 2021 (#31!)

Thank you for watching! Please Subscribe! Content Links: MoCoV3: https://arxiv.org/pdf/2104.02057.pdf Revisiting Simple Neural Probabilistic Language Models: https://arxiv.org/pdf/2104.03474.pdf Large-scale forecasting: Self-supervised learning framework for hyperparameter tuning: https:/

From playlist AI Research Weekly Updates