In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε2 = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra. In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigid transformation has six real degrees of freedom, three for translations and three for rotations, dual quaternions obeying two algebraic constraints are used in this application. Similar to the way that rotations in 3d space can be represented by quaternions of unit length, rigid motions in 3d space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics (see McCarthy), and in applications to 3D computer graphics, robotics and computer vision. (Wikipedia).
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
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 Lie-algebra of Quaternion algebras and their Lie-subalgebras
In this video we discuss the Lie-algebras of general quaternion algebras over general fields, especially as the Lie-algebra is naturally given for 2x2 representations. The video follows a longer video I previously did on quaternions, but this time I focus on the Lie-algebra operation. I st
From playlist Algebra
Quaternion algebras via their Mat2x2(F) representations
In this video we talk about general quaternion algebras over a field, their most important properties and how to think about them. The exponential map into unitary groups are covered. I emphasize the Hamiltionion quaternions and motivate their relation to the complex numbers. I conclude wi
From playlist Algebra
Dual basis definition and proof that it's a basis In this video, given a basis beta of a vector space V, I define the dual basis beta* of V*, and show that it's indeed a basis. We'll see many more applications of this concept later on, but this video already shows that it's straightforwar
From playlist Dual Spaces
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
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
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
GAME2020 0. Steven De Keninck. Dual Quaternions Demystified
My GAME2020 talk on PGA as an algebra for the Euclidean group. Follow up on my SIGGRAPH 2019 talk : https://youtube.com/watch?v=tX4H_ctggYo More info on https://bivector.net
From playlist Bivector.net
QED Prerequisites Geometric Algebra 24 Paravectors
In this lesson we discover yet another way to partition the components of a general multivector. In this method, the partitioning is entirely dependent on a choice of reference frame. That is \gamma_0 must be chosen and it represents the 4-velocity of an observer who is stationary in that
From playlist QED- Prerequisite Topics
Siggraph2019 Geometric Algebra
**Programmer focused part** starts at 18:00 Try the examples here https://enkimute.github.io/ganja.js/examples/coffeeshop.html The Geometric Algebra course at Siggraph 2019. Intro : Charles Gunn (00:00 - 18:00) Course : Steven De Keninck (18:00 - end) Course notes, slides, software, disc
From playlist Bivector.net
Geometric Algebra - Duality and the Cross Product
In this video, we will introduce the concept of duality, involving a multiplication by the pseudoscalar. We will observe the geometric meaning of duality and also see that the cross product and wedge product are dual to one another, which means that the cross product is already contained w
From playlist Geometric Algebra
Representation theory: The Schur indicator
This is about the Schur indicator of a complex representation. It can be used to check whether an irreducible representation has in invariant bilinear form, and if so whether the form is symmetric or antisymmetric. As examples we check which representations of the dihedral group D8, the
From playlist Representation theory
SIGGRAPH 2022 - Geometric Algebra
The SIGGRAPH 2022 course on Geometric Algebra. by Alyn Rockwood and Dietmar Hildenbrand
From playlist Introductory
A Mathematical Introduction to 3d N = 4 Gauge Theories (Lecture 1) by Mathew Bullimore
PROGRAM : QUANTUM FIELDS, GEOMETRY AND REPRESENTATION THEORY 2021 (ONLINE) ORGANIZERS : Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pan
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Nigel Hitchin "Higgs bundles, past and present" [2012]
2012 FIELDS MEDAL SYMPOSIUM Thursday, October 18 Geometric Langlands Program and Mathematical Physics Nigel Hitchin, Oxford University Higgs bundles, past and present The talk will be an overview of the moduli spaces of Higgs bundles, or equivalently solutions to the so-called Hitchin eq
From playlist Number Theory
Dr Leo Dorst' Keynote talk at CGI2020
A high-speed introduction to the Algebra of planes.
From playlist Bivector.net
Set Theory (Part 14c): More on the Quaternions
No background in sets required for this video. In this video, we will learn how the quaternions can be thought of as pairings of complex numbers. We also will show how the quaternions can be written as a 2x2 complex matrix as opposed to a 4x4 real matrix and how the unit quaternions form t
From playlist Set Theory