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: 8864
Mathoma

Hand calculation of a quaternion rotation of space based on the Lattice Method

Views: 6429
silencedidgood

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: 134
David Parker

How both cross-product and dot-product of three-dimensional vectors come from the multiplication of four-dimensional numbers (quaternions)

Views: 139
Martin Thomas

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: 14051
MathDoctorBob

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: 47565
Mathoma

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: 15748
Mathoma

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/
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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: 557469
3Blue1Brown

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: 182538
3Blue1Brown

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: 14235
silencedidgood

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: 2088
Doug Sweetser

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: 16721
Jorge Rodriguez

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: 27484
Mathoma

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: 413
David Parker

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: 130216
Daniel Finlay

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: 2647
Mathoma

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: 11641
UniAdel

So I finally decided to do a video on quaternions and slerping. This is mainly for educational purposes as using the built in lerp method will already slerp the rotation between two cframes.
Here are the links I mentioned throughout the video:
- 2D rotation matrix: http://wiki.roblox.com/index.php?title=2D_Collision_Detection#Prerequisites_to_Method_2:_Corners_of_a_rotated_shape
- Rodrigues' rotation formula: https://www.youtube.com/watch?v=OKr0YCj0BW4
- Rotation matrix to quaternion: http://wiki.roblox.com/index.php?title=Quaternions_for_rotation#Quaternion_from_a_Rotation_Matrix
- Slerp module: https://github.com/EgoMoose/ExampleDump/blob/master/Scripts/slerp.lua
If you have a suggestion please leave a comment as this was my first time making a video like this so for the future your feedback helps!
Thanks for watching!

Views: 29794
EgoMoose

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: 73973
Jorge Rodriguez

Virtual Reality by Prof Steven LaValle, Visiting Professor, IITM, UIUC. For more details on NPTEL visit http://nptel.ac.in

Views: 6164
nptelhrd

Views: 624
jae kim

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: 5582
Jorge Rodriguez

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: 1096
silencedidgood

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: 124
David Parker

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: 1617
J Zelawin

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: 531
David Metzler

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: 274432
3Blue1Brown

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: 27658
njwildberger

Accompanying video for paper "Skinning with Dual Quaternions", I3D 2007.

Views: 16325
Ladislav Kavan

A poem about Quaternions:
A quaternion is like a vector, but with a "w"
To construct one, use an axis and an angle, that's what we do
For rotations it must be normal, or otherwise its pure
So we normalise, divide by length, just to be sure
To invert a normal quaternion, we negate x, y and z
Multiply quaternion, vector, inverse quaternion and it rotates don't you see
A rotation of 0 radians is the same as two pi
To convert a quaternion to a matrix, we use the API
So here's a health to old Hamilton, your inventor it would appear
And to imaginary numbers floating in the hypersphere
- Dr Bryan Duggan

Views: 18106
Bryan Duggan

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: 3230
silencedidgood

Humane Rigging is the 8th DVD in the Blender Open Movie Workshop series!
The entire contents of this video is © 2012 by Nathan Vegdahl and The Blender Foundation, and is licensed under CC BY 3.0
The music used in various videos on this playlist is by DanoSongs.com, and is licensed under CC BY 3.0

Views: 48296
Sutrabla

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: 167406
Mathoma

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: 134438
Unity

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: 15180
wolframmathematica

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: 183440
Unity

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 connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
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.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 15288
njwildberger

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: 4838
smog83

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: 2740
Doug Sweetser

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: 1799
Mathoma

In this second part, we'll investigate the algebraic properties and geometric meaning of this extended dot and wedge product. We'll see that vectors in the plane of a bivector anticommute with the bivector under the geometric product and vectors orthogonal to the plane of a bivector will commute with that bivector. This will allow us to say in general what it means to dot a vector with a bivector and what it means to wedge a vector with a bivector.
References / Further Reading:
1. Lasenby and Doran's "Geometric Algebra for Physicists".
2. Recommended video on Clifford algebra:
https://www.youtube.com/watch?v=rqadK8-dN-Y
3. And other video series covering Geometric Algebra (Nick Okamoto): https://www.youtube.com/watch?v=P-IKemH3jsg&list=PLQ6JJNfj9jD_H3kUopCXkvvGoZqzYOzsV

Views: 2561
Mathoma

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: 15848
njwildberger

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: 1720
Mathoma

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
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Game Math Theory is a series intended to give a solid understanding of the mathematical concepts that underlie the game development process.
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♪ Baby Plays Electro Games
http://teknoaxe.com/Link_Code_2.php?q=326

Views: 75595
Brackeys

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: 905
silencedidgood

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: 652266
GuerrillaCG

In this video, we will talk about the dual numbers and see how they are quite similar to other number systems like the complex numbers and the split complex numbers. In particular, these numbers work similar to infinitesimals and therefore give rise to derivatives quite naturally.

Views: 5519
Mathoma

Graphical animation realized in "Processing" . 6 vectors in 3d space rotation around each other. Using mathematician NJ Wildberger's rotation formula with quaternions. Silent movie.

Views: 457
Vilbjørg Broch

In this video, we make the camera rotate using quaternions, and set up the camera transformation matrix.
Code: https://github.com/BennyQBD/3DGameEngine

Views: 18871
thebennybox