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Quaternions: Extracting the Dot and Cross Products
 
18:49
The most important operations upon vectors include the dot and cross products and are indispensable for doing physics and vector calculus. The dot product gives a quick way to check whether vectors are orthogonal and the cross product calculates a new vector orthogonal to both its inputs. These vector operations were originally derived from the analysis of quaternions and eventually parted ways with quaternions to become our modern vector analysis. Much bickering ensued in the infancy of vector analysis, with the big names including Hamilton, Gibbs, and Heaviside, but we will focus on the math here. In this video, I will show you how the dot and cross product arise from dissecting the formula for quaternion multiplication and allows us to write quaternion multiplication using dot and cross products. In addition, I will show the converse, how the dot and cross products can be defined using quaternion multiplication.
Views: 8244 Mathoma
How to calculate Quaternion dot product - ProgrammingTIL #186 3D Math 30 tutorial video screencast
 
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Subscribe! http://bit.ly/subdavidwparker In this episode, I discuss how to calculate the quaternion dot product. Sign up for my Newsletter: https://www.programmingtil.com/ Follow me on Twitter: https://twitter.com/davidwparker and https://twitter.com/programmingtil Concepts: * Quaternion dot product Resources: * Code / Slides: https://github.com/davidwparker/programmingtil-3d-math/tree/master/0030-quaternions-diff-dot * Slides JS: https://github.com/hakimel/reveal.js/ * Math JS: https://www.mathjax.org/ Current Schedule: Monday / Wednesday / Friday - 3D Math or WebGL
Views: 106 David Parker
Cross products, dot products, and quaternions
 
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How both cross-product and dot-product of three-dimensional vectors come from the multiplication of four-dimensional numbers (quaternions)
Views: 125 Martin Thomas
Quaternion Rotation and the Sandwich Product
 
13:32
Hand calculation of a quaternion rotation of space based on the Lattice Method
Views: 6390 silencedidgood
3D Rotations in General: Rodrigues Rotation Formula and Quaternion Exponentials
 
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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 of view. Using quaternions allows us to write a very compact formula which will be familiar to those who have used quaternions to do rotations.
Views: 45829 Mathoma
What are quaternions, and how do you visualize them?  A story of four dimensions.
 
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How to think about this 4d number system in our 3d space. Thanks to supporters: https://www.patreon.com/3blue1brown With extra-special thanks here: http://3b1b.co/quaternion-thanks Quanta article on quaternions: https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/ The math of Alice in Wonderland: https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/ Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
Views: 492976 3Blue1Brown
Math for Game Developers - Rotation Quaternions
 
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We build on the idea of axis-angle rotations to start constructing quaternions. Find the source code here: https://github.com/BSVino/MathForGameDevelopers/tree/quaternions Question? Leave a comment below, or ask me on Twitter: https://twitter.com/VinoBS
Views: 71927 Jorge Rodriguez
Quaternions - Unity Official Tutorials
 
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Watch this video in context on Unity's learning pages here - http://unity3d.com/learn/tutorials/modules/intermediate/scripting/quaternions Quaternions are a system of rotation that allowed for smooth incremental rotations in objects. In this video, you were learn about the quaternion system used in Unity and you will explore a few of the methods that allow you to work with it. Help us caption & translate this video! http://amara.org/v/V69K/
Views: 129291 Unity
Quaternions - identity, magnitude, conjugate, inverse ProgrammingTIL #184 3D Math 28 tutorial video
 
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In this episode, I discuss quaternions. Specifically, identity, magnitude, conjugate, and inverse for quaternions. Sign up for my Newsletter: https://www.programmingtil.com/ Follow me on Twitter: https://twitter.com/davidwparker and https://twitter.com/programmingtil Concepts: * Quaternions - identity, magnitude, conjugate, inverse Resources: * Code / Slides: https://github.com/davidwparker/programmingtil-3d-math/tree/master/0028-quaternions-iden-mag-conj-inv * Slides JS: https://github.com/hakimel/reveal.js/ * Math JS: https://www.mathjax.org/ Current Schedule: Monday / Wednesday / Friday - 3D Math or WebGL
Views: 309 David Parker
Math for Game Developers - Multiplying Quaternions
 
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We learn how to combine two rotation quaternions to make one quaternion that does both rotations. Derivation of the quaternion multiplication in this video can be found in the book "3D Math Primer For Graphics And Game Development" by Fletcher Dunn and Ian Parberry The source code for this video will be combined with the source code for the next video. Question? Leave a comment below, or ask me on Twitter: https://twitter.com/VinoBS
Views: 16120 Jorge Rodriguez
Quaternion Multiplication - ProgrammingTIL #185 3D Math 29 tutorial video screencast
 
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Subscribe! http://bit.ly/subdavidwparker In this episode, I discuss how to multiply two quaternions together. Sign up for my Newsletter: https://www.programmingtil.com/ Follow me on Twitter: https://twitter.com/davidwparker and https://twitter.com/programmingtil Concepts: * Quaternions * Quaternion multiplication Resources: * Code / Slides: https://github.com/davidwparker/programmingtil-3d-math/tree/master/0029-quaternions-multiply * Slides JS: https://github.com/hakimel/reveal.js/ * Math JS: https://www.mathjax.org/ Current Schedule: Monday / Wednesday / Friday - 3D Math or WebGL
Views: 306 David Parker
04 3D Bouncy Ball 05 Quaternion Rotation
 
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Blender is a free open source 3D animation software available at https://www.blender.org/download/ If you enjoy using Blender please donate at https://www.blender.org/foundation/donation-payment/ If you find them helpful please purchase the DVD at https://store.blender.org/product-category/training/ Thank You. As stated in Part 1 of this Blender DVD training: Humane Rigging video series these videos are released under the Creative Commons and are free to use and copy. The music used in various videos on this DVD is by DanoSongs.com and is licensed under CC BY 3.0
Views: 1378 J Zelawin
Quaternions Explained by Dan
 
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An overview of what quaternians are, how to do a basic rotation in 3d space, and how to use software to do it easier. Made because I thought I worked harder than I should've needed to to rotate things in 3D space myself! If you're trying to do quaternion arithmetic yourself, my favorite guide is here: http://www.youtube.com/watch?v=r9jWCbpLvHw It involves lattice multiplication, so you'd better be prepared! Khanacademy has a great primer here: https://www.khanacademy.org/math/arithmetic/multiplication-division/lattice_multiplication/v/lattice-multiplication And for a more detailed introduction, this much more personable teacher has appeared on the youtube scene since I made this video, and I've heard great things about it: http://www.youtube.com/watch?v=uRKZnFAR7yw Good luck!
Views: 129130 Daniel Finlay
Quaternions as 4x4 Matrices - Connections to Linear Algebra
 
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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 already know how to do more or less any calculation with quaternions. Moreover, we will see that the fundamental relations between the quaternionic imaginary units, i, j, and k also hold in matrix form. If you enjoyed this video, please subscribe and leave comments!
Views: 14735 Mathoma
#44 3D Game Engine Tutorial: Quaternions & Transformation
 
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In this video, we update our transform class to use quaternion rotations. Code: https://github.com/BennyQBD/3DGameEngine
Views: 6151 thebennybox
Quaternion code part 4: inverse, difference, dot product ProgrammingTIL #217 3D Math 52 tutorial
 
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Subscribe! http://bit.ly/subdavidwparker In this episode, I discuss more Quaternion code. We implement inverse, difference, and dot product Sign up for my Newsletter: https://www.programmingtil.com/ Follow me on Twitter: https://twitter.com/davidwparker and https://twitter.com/programmingtil Concepts: * Quaternion inverse * Quaternion difference * Quaternion dot product Resources: * https://github.com/davidwparker/programmingtil-3d-math/tree/master/0052-code-quat-4 Current Schedule: Monday / Thursday - 3D Math or WebGL
Views: 104 David Parker
The Vector Algebra War
 
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There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs’ three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of the primary goals of nineteenth century science was to suitably describe vectors in three-dimensional space. This situation has also had the unfortunate consequence of fragmenting knowledge across many disciplines, and requiring a significant amount of time and effort in learning the various formalisms. We thus historically review the development of our various vector systems and conclude that Clifford’s multivectors best fulfills the goal of describing vectorial quantities in three dimensions and providing a unified vector system for science.
Views: 10493 UniAdel
Example of Quaternions
 
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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.
Views: 13788 MathDoctorBob
Quaternions and 3d rotation, explained interactively
 
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Go experience the explorable videos: https://eater.net/quaternions Ben Eater's channel: https://www.youtube.com/user/eaterbc Previous video on Quaternions: https://youtu.be/d4EgbgTm0Bg Special thanks to these supporters: http://3b1b.co/quaternion-explorable-thanks Nice explanation of Gimbal Lock: https://youtu.be/zc8b2Jo7mno Great videos comparing Euler angles and quaternions, from the perspective of an animator: https://youtu.be/syQnn_xuB8U https://youtu.be/4mXL751ko0w Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
Views: 157290 3Blue1Brown
GEM Blog 29: Deriving the Maxwell Source Equations Using Quaternions (2/5)
 
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This is a dramatic reading of "Deriving the Maxwell Source Equations Using Quaternions", a blog by the Stand-Up Physicist available at Science20.com. Start with the big picture - a definition of what the Maxwell equations do, account for all spacietime changes of spacetime change of a spacetime potential caused by a current density mediated by massless photons who have no sense of their own history. The rules for multiplying quaternions are explained. The simplest quaternion derivative of a potential contains both the electric field E and magnetic field B. The product of the two ways to write the quaternion derivative of a potential generates the Lagrange density for the Maxwell source equations, B squared minus # squared. Much work goes into writing that out component by components, all 22 terms. The Euler-Lagrange equations are then used to generate both Gauss's and Ampere's Laws. A visualization of the two laws allows for a comparison. The weak and strong forces are profoundly similar to EM, the key difference being the symmetry groups involved. Because quaternions have both multiplication and division, it is possible to write a form of Gauss's and Ampere's laws that have both SU(2) and SU(3) symmetries with underlies the weak and the strong forces. Whether that is at all relevant to Nature is an open question.
Views: 2679 Doug Sweetser
Hand Calculation of Quaternion Rotation
 
19:48
Using a simple rotation to prove the sandwich product calculation of quaternion multiplication. A point described as (1,0,0) is rotated 30 degrees about the z axis ending up of course at (cos(30),sin(30),0)
Views: 13948 silencedidgood
Set Theory (Part 14b):  Quaternions and 3D Rotations
 
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No background in sets needed for this video - learn about the foundations of quaternions, derivation of the Hamilton product, and their application to 3D rotations. We will also see how dot and cross products are related to quaternion math. This video will be of particular interest to computer programmers and physicists. Please leave your thoughts and comments below!
Views: 2570 Mathoma
Understanding Quaternions through Geometric Algebra
 
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The book I'm learning from: http://geometricalgebra.net My Twitter: http://twitter.com/VinoBS Errata: 17:40 This development of contraction product assumes that a and b are orthogonal to begin with. With two arbitrary vectors a and b that are not necessarily orthogonal, you don't have that a|X = b. 51:30 You'll see theta degrees of rotation if a and b are theta/2 degrees apart. 53:03 Hint: (ab)^-1 = b^-1 a^-1
Views: 5187 Jorge Rodriguez
Overview (Video 2/14).
 
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Note: If you take the tensor product over C between the complex quaternions and the complex octonions, then you arrive at RxCxHxO. Video 2 of 14 Series: Division algebras and the standard model Some short videos filmed by Vincent Lavigne Seminar by C. Furey, Walter Grant Scott Research Fellow in Physics Trinity Hall, University of Cambridge http://damtp.cam.ac.uk/people/nf252/ https://arxiv.org/pdf/1611.09182.pdf https://arxiv.org/abs/1603.04078 https://arxiv.org/abs/1405.4601 https://www.furey.space
Views: 20702 Cohl Furey
The rotation problem and Hamilton's discovery of quaternions (II) | Famous Math Problems 13b
 
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This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations. This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science! Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
Views: 27113 njwildberger
Three views of the vector triple product (Part 15)
 
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Here I quickly go through a fourth "bonus" view of the vector triple product, introducing quaternions and showing how the vector triple product identity can be used to show that quaternion multiplication is associative.
Views: 528 David Metzler
Hand calculation of a quaternion rotation about a 60 degree line
 
15:00
This video shows how to use the sandwich product to rotate a point (1,0,0) about an axis defined by the line in the x-y plane that is commonly described as a 60 degree angle from the axis. The point is rotate coincidentally by 60 degrees. Of course we all should remember that 60 degrees is Pi/3.
Views: 3203 silencedidgood
Multiplying two Quaternions by Hand
 
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This shows the second half of the "sandwich" product calculations used in quaternion rotations of points. This just shows the mechanics of an operation that will undoubtedly be done with a computing machine
Views: 1047 silencedidgood
quaternion vs euler rotation
 
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yellow arrow - quaterion rotation. red arrow - euler rotation.
Views: 16481 SoloDev
Vector Maths - Unity Official Tutorials
 
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Watch this video in context on Unity's learning pages here - http://unity3d.com/learn/tutorials/modules/scripting/lessons/vector-maths-dot-cross-products A primer on Vector maths - as well as information on the Dot and Cross products.
Views: 179078 Unity
Euler (gimbal lock) Explained
 
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In this video we explore Euler Rotations, the most common method for orienting objects in 3d. It's by-product "gimbal lock" can cause headaches for animators because the animated motion can move in strange ways. Here we learn how euler's "rotation order" is a bit like hierachies, and how changing this order can help us to avoid gimbal problems. This is demonstrated with a solution to a common camera problem, by finding the correct rotation order.
Views: 618815 GuerrillaCG
Geometric Algebra - Duality and the Cross Product
 
01:19:29
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 within geometric algebra when working in three dimensions. It should also be clear, once we reflect upon duality, why it only makes sense to speak of a cross product in three dimensions. We will finally go through a number of applications of duality such as the equivalence of rotating about an axis and through a plane in G(3), the quaternionic product, the vector-bivector product, the scalar triple product, and the vector triple product. Geometric Algebra playlist: https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K References / Further Reading: 1. Lasenby and Doran's "Geometric Algebra for Physicists". https://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran/dp/0521715954 2. "A Survey of Geometric Algebra and Geometric Calculus" by Alan Macdonald: http://www.faculty.luther.edu/~macdonal/GA&GC.pdf 3. "Synopsis of Geometric Algebra" and "Geometric Calculus" by David Hestenes: http://geocalc.clas.asu.edu/html/NFMP.html Music: J.S. Bach's Concerto for Two Violins in D minor, 1st Mov.
Views: 1328 Mathoma
Quaternions Explained Briefly
 
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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 quaternion multiplication. For the sake of brevity, I don't cover the famous application to 3D rotations in this video (perhaps in a subsequent one) but, of course, one must first know how to multiply two quaternions before talking about specific applications.
Views: 161097 Mathoma
From Quaternion to 3D Rotation
 
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http://demonstrations.wolfram.com/FromQuaternionTo3DRotation The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. Any nonzero quaternion ? has a corresponding unitary (length one) quaternion in the same direction as ?. Unitary quaternions are an elegant and efficient way to formalize 3D rotations. Contributed by: Isabelle Cattiaux-Huillard and Gudrun Albrecht Audio created with WolframTones: http://tones.wolfram.com
Views: 14945 wolframmathematica
Cross products in the light of linear transformations | Essence of linear algebra chapter 11
 
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For anyone who wants to understand the cross product more deeply, this video shows how it relates to a certain linear transformation via duality. This perspective gives a very elegant explanation of why the traditional computation of a dot product corresponds to its geometric interpretation. *Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result since the determinant is unchanged after a transpose, but given how I've framed most of this series I think it is more intuitive to go with a column-centric approach. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown
Views: 249301 3Blue1Brown
Quaternion Slerp v.s. Matrix Lerp
 
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This is the video that shows different results when you using Quaternion slerp and Matrix lerp on XNA Framework. http://blogs.msdn.com/ito/archive/2009/05/01/more-bones-04.aspx ひにけにXNAの記事用に作った動画です。
Views: 50675 higeneko
Euler's Formula for the Quaternions
 
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In this video, we will derive Euler's formula using a quaternion power, instead of a complex power, which will allow us to calculate quaternion exponentials such as e^(i+j+k). If you like quaternions, this is a pretty neat formula and a simple generalization of Euler's formula for complex number exponentials.
Views: 26774 Mathoma
Hand Calculation of a Quaternion rotation about a 60 degree line - Part 2
 
14:56
This video shows how to use the sandwich product to rotate a point (1,0,0) about an axis defined by the line in the x-y plane that is commonly described as a 60 degree angle from the axis. The point is rotate coincidentally by 60 degrees. Of course we all should remember that 60 degrees is Pi/3.
Views: 897 silencedidgood
Quaternions
 
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Virtual Reality by Prof Steven LaValle, Visiting Professor, IITM, UIUC. For more details on NPTEL visit http://nptel.ac.in
Views: 5642 nptelhrd
Geometric Algebra - Rotors and Quaternions
 
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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 of geometric algebra. We will also see that the rotors form a group with the geometric product. Geometric Algebra playlist: https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K References / Further Reading: 1. Lasenby and Doran's "Geometric Algebra for Physicists". https://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran/dp/0521715954 2. "A Survey of Geometric Algebra and Geometric Calculus" by Alan Macdonald: http://www.faculty.luther.edu/~macdonal/GA&GC.pdf 3. "Synopsis of Geometric Algebra" and "Geometric Calculus" by David Hestenes: http://geocalc.clas.asu.edu/html/NFMP.html Note: 1. At 22:15, supposing that i is not equal to j. Music: J.S. Bach's Concerto for Two Violins in D minor, 1st Mov.
Views: 1449 Mathoma
Math for Game Developers - Quaternion Inverse
 
05:24
We learn how to construct the inverse quaternion that rotates in the opposite direction. Find the source code here: https://github.com/BSVino/MathForGameDevelopers/tree/quaternion-inverse Question? Leave a comment below, or ask me on Twitter: https://twitter.com/VinoBS
Views: 13315 Jorge Rodriguez
Skinning with Dual Quaternions
 
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Accompanying video for paper "Skinning with Dual Quaternions", I3D 2007.
Views: 16006 Ladislav Kavan
Math for Game Developers - Slerping Quaternions
 
08:32
Spherical Linear Interpolation of Quaternions. Find the source code here: https://github.com/BSVino/MathForGameDevelopers/tree/quaternion-slerp Question? Leave a comment below, or ask me on Twitter: https://twitter.com/VinoBS
Views: 11323 Jorge Rodriguez
quaternions
 
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With & without implementing dot product estimation
Views: 28 aswinsiva91
Game Math Theory - VECTORS
 
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This video briefly explains what Vectors are and how they can be used in a game. ♥ Support my videos on Patreon: http://patreon.com/brackeys/ ● Read more about Vectors: https://www.mathsisfun.com/algebra/vectors.html ···················································································· Game Math Theory is a series intended to give a solid understanding of the mathematical concepts that underlie the game development process. ···················································································· ♥ Subscribe: http://bit.ly/1kMekJV ● Website: http://brackeys.com/ ● Facebook: https://facebook.com/brackeys/ ● Twitter: https://twitter.com/BrackeysTweet/ ········································­­·······································­·­···· ► All content by Brackeys is 100% free. I believe that education should be freely available to everyone. Want to support me in my mission? Become a Patron: http://patreon.com/brackeys/ ········································­­·······································­·­···· ♪ Baby Plays Electro Games http://teknoaxe.com/Link_Code_2.php?q=326
Views: 70620 Brackeys
Polar Representations of Quaternions
 
08:04
The polar representation of quaternions is identical to what happens for complex numbers except the imaginary i gets replaced by an imaginary 3-vector, i, j, and k. To get to an event from the origin requires heading off in a certain direction at a certain speed. This is what the polar representation of quaternions accomplishes. The preprint is available at http://VisualPhysics.org/preprints/QMN1009.2352
Views: 2077 Doug Sweetser
the quaternion handshake
 
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You had the quaternion in you hands all this time and didn't know about it! the quaternions are mathematical abstract group/ ring / algebra that can be used to represent axial rotations. http://en.wikipedia.org/wiki/Quaternion
Views: 4826 smog83
The rotation problem and Hamilton's discovery of quaternions III | Famous Math Problems 13c
 
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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 remarkable structure. A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play. We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems! In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!! Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
Views: 15488 njwildberger
Quaternions, spatial Pythagorean hodographs, and rotations in three and four dimensions
 
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Title: Quaternions, spatial Pythagorean hodographs, and rotations in three and four dimensions Speaker: Professor Rida T. Farouki Date: November 29, 2012 Series: UCD MAE Weekly Seminar Where: Department of Mechanical and Aerospace Engineering, University of California, Davis. Abstract: Quaternions, the first example of a non-commutative algebra, arose as a by-product of Hamilton's failed attempt to construct an "algebra of triples". Hamilton envisaged the quaternions as the "new language" of science and technology, but their place was usurped by vector analysis, an algebraically crude and overtly pragmatic subset of the quaternion algebra. A simple quaternion expression generates Pythagorean quartuples of polynomials, yielding an elegant rotation-invariant characterization for spatial Pythagorean-hodograph curves. Quaternions provide compact, intuitive descriptions for 3-dimensional rotations, a fact that has generated a renewed interest in them for applications in animation, robotics, computer graphics, and related fields. Quaternions can also describe 4-dimensional rotations, whose strange properties provide a cautionary tale against extrapolating our geometric intuition to higher-dimension Euclidean spaces.
Views: 1333 Jason Moore

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