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- เผยแพร่เมื่อ 1 ก.ค. 2024
- 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.
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"Now oftentimes in math, we have many different ways of viewing the same thing."
Bingo!!
One of the ways I like to define the quaternions, is as complex numbers over the complex numbers, where the 2 imaginary units are distinct, and anticommute.
If we define
q = A + Bj
where
A = a + bi, B = c + di
you then have
q = a + bi + cj + dij
So then you just call this new, "product" of imaginary units, a new, third imaginary unit, k,
q = a + bi + cj + dk
And now, all the multiplication rules for imaginary units can be worked out. From our premises that i² = j² = -1, and ij = -ji = k, we have
ij = -ji = k
jk = jij = -ij² = i
ki = iji = -i²j = j
ik = iij = -j
kj = ijj = -i
k² = ijij = -iijj = -1
Fred
@Mathoma when multiplying both side by a variable is it a rule that, that variable comes first. Eg.. j*j*k=j*i vs j*k*j=i*j or is this only the case because were talking about higher dimensions here? It is it dependent on the side of the variable which is being multiplied?
%
I haven't finished watching the video yet, but I came down to thank you for using black background and white text. EVERYONE SHOULD DO LECTURES THIS WAY!!! My eyes are not burning for once holy shit dude.. THANK YOU
Me thinks NOT!
Spoken like someone watching at 3am. Also me
Hello,
Many thanks indeed for uploading this set of five videos about quaternions. Most interesting. I'm 63 and now at last I know about quaternions that were discovered by Hamilton an Irish mathematician as you know. I live not too far away from the place a canal where the plaque with his formulation of quaternions is written at the spot where it all became clear to Hamilton while he was walking on his way to his university Trinity College Dublin from where he lived that was what is Dunsink Observatory today and then.
All the best,
Peter Nolan.(Ph.D., experimental physics). Dublin. Ireland.
+Peter Nolan
Even though people don't often learn of quaternions anymore, the concepts live on whenever you write a dot product or cross product. You might be interested in geometric algebra, too (or Clifford algebra more generally). I'm putting together videos on that topic but when I eventually start talking about the geometric algebra of 3D, the quaternions will pop up yet again, this time with a different interpretation than "arrow" or "vector".
Hello,
I am just about to watch all of your videos a second time starting with the first one above. The astronomer I was telling you about above was telling me that quaternions are used to steer satellites and as you undoubtedly know there are many other applications for them as well.
The i, j and k in quaternions are not the same as the i, j and k that we used when we were being taught about vectors starting in secondary school and I found that a bit confusing to start with.
All the best and many thanks,
Peter Nolan.(Ph.D., experimental physics). Dublin. Ireland.
+Peter Nolan
Yeah, they mean different things but those i,j,k are nice historical artifacts from the quaternion days of physics. Of course, nowadays they are just placeholders for x,y,z with no algebraic significance.
Many thanks for that clarification. I had not heard the word "placeholder" before. As they say in America every day is a day at school. I'm 63.
All the best,
Peter Nolan.(Ph.D., experimental physics). Dublin. Ireland.
+Peter Nolan
Perhaps "placeholder" wasn't a good way to describe the current function of i,j,k now that I think of it. They certainly represent the basis vector in the x,y,z directions so they do indicate something.
Thanks so much you really saved me. I could not find anyone who could explain this topic better than this
This is how beautiful a concept can be explained! Flawless, thanks a lot :)
This was wonderfully explained. I cant tell you how many channels ive been thru trying to find something like this.
Many try to flex their knowledge of the topic without realising that some of their viewers are there to learn the topic from the ground up.
Amazing work!
I find the quaternion multiplication table very informative, as well as a refresher from the other parts of your video. Thanks!
I still learn very (relatively speaking) math, in fact I have only briefly touched complex numbers and i, yet this video was very intresting and taught me some basic quaterion rules. Now I probably won't use this knowledge for another few semesters but videos like these always keep me motivated to learn more and get there!
What college you at? I like Harvard because math 55 teaches me about topology, where there are subsets, elements, n and k cells, intersection area of, total area of, neighborhoods, hausdorff neighorhoods, euclidean space, hausdorff space, converting topology into metric, paths, quantifications, including uniqueness quantification, existential quantification, and universal quantification, closures, interiors, boundaries, CW Complex, isotopy, homotopy, homomorphism, morphing into other shapes, paths, congruent, changed to, if and only if, functors, open balls, closed balls, fibers, and more.
You think and explain in the way I do it. Kudos sir! And thank you for your effort!
That's the first time I actually understood what Quaternions are all about. Kudos to you!
Thank You! That's the only explanation of Quaternions that I could understand properly, very well explained
No angry comments, the opposite, I am happy to see and listen your awesome exposition. Thank you very much Mathoma.
Thanks for the video, I'm currently working on a robotics project and it's the first time I heard about quaternions, your explanation was pretty spot-on as an introduction to the topic! Also your channel seems really interesting, great job! Greetings from the other side of the wall!
I like how you write the letter 'q'. Also good explanation.
Thank you very much for explaining quaternions in a very simple way. It helped me a lot.
I have a presentation to do in my first year of university (i'm doing a double degree in maths and physics) and we chose quaternions with my two friends. I'm the only one who speaks English so I'll be able to brag and make them think I'm a quaternion expert because I watched your video. It was really well-explained and easy to understand, thank you !
OMG OMG OMG!! I was stuck reading this topic in a book and you finally has given to me the final understanding about this topic! You are amazing dude
What a nice and clear explanation, thank you!
Thank you for this video ! I'm currently in high school and I'm developing some video games. Since I didn't know how quaternions worked I was using Euler's angles which are (magically) translated to rotation by some modern game engine (i.e. Unity3D). Now I'll be able to work directly with quaternions so thanks for quenching my thirst of curiosity !
+Black Rainbow
You're welcome; quaternions are one of my favorite topics in math so I always enjoy talking about them. This particular video probably won't tell you much on why they work in rotations, but hopefully that will make sense when you see my other quaternion videos. The more I study this topic, the more I'm convinced Euler angles and other contortionist routines using matrices are the wrong way to think of 3D rotations. Quaternions and more generally the geometric algebra of R^3 are too natural and the formulas are too concise for them to not be the best way to understand rotation.
Very nice explanation, especially liked the bit about how an abstract mathematical idea became an useful one.
You are a legend ! Thanks for a wonderful lecture
Love the "blackboard" presentation. Great video 👍
My math only goes up to abstract algebra, so this was a nice intro for me, thanks
I liked the video. It leaves many things un-answered, but it it useful. It is just natural that one should read and search more on the topic before one understands what is going on.
A brilliant explanation of quaternion algebra. William Rowan Hamilton would surely be impressed. Well done.
Thanks for this introduction to quaternions!
This is the most simple genious idea ever. 👌
Sir your content is amazing. Thank you so much.
You are excellent in explaining it. Thanks for the effort
Thank you for your nice explanation. Which book(s) did you use in this series on quaternions? Or which book(s) would you recommend?
I'd like to see more applications of quaternions; in gyroscopes for example. Nice and clear lecture, thanks!
I LOVE THIS CHANNEL!!!!
This was soooooooooo awesome!
very well explained. you have just gained a subscriber.
Great explanation, congrats!
Khan Academy 2.0! This is great!
+Arongil Productions
Yeah, his videos are some of my main inspirations, except I work at a slightly higher level.
Mathoma khan academy is a little bit too basic even for beginners
Thank you for the explanation and understanding of the Hamilton graffiti.
at 1:31 when you explain so simply the R^2, you have my like on the video
I was with you Doc all the way until you got here: 11:00
When you multiplied by ijk = -1
=> iijk = -I... All these calculations take graduating layers of abstract thought and my thoughts failed after the third level of complexity. I love math but I could never get beyond algebra 1 to precalculus because so many assumptions of the abstract seem incorrect.
I don’t know if that makes sense, but hopefully we can help a whole new generation break the fear of beyond. Live long and prosper sir!
🖤💪🏽👌
Ok, I'm replying to a two year old comment but I paused at the very same point. This is me gathering my thoughts - it helps me to write it down - so just ignore me.
The issue is that normally -1 * -1 = 1. So vec3(-1, -1, -1) * vec3(-1, -1, -1) = vec3(1, 1, 1). Which is the same as vec3(1, 1, 1) * vec3(1, 1, 1) when obviously it's not the same. I mean, -1 * -1 = 1 is just a definition but it's a crappy one in this context because it's asymmetric. Assumptions of the abstract as you say. Just because a vector is pointing in the negative direction doesn't mean we want it to behave differently when we multiply it. Obviously we don't.
So let's just change that rule. So now, when it's negative, let's multiply its positive and then add the negative sign back after the multiplication is done and return that result. Do this and -1 * -1 now outputs -1. I think proper maths takes the sqrt(-1) but the result is the same so who cares?
I'm not sure, but I suspect all this mathematical hocus-pocus is just to make negative numbers behave like positive numbers and not magically flip on you. In computer terms you'd call this patching a bug. Clearly, -1 * -1 shouldn't =1, at least in this context.
You're great, thanks for this great explanation
Thanks man, these are really helpful!
I AM ANGRY THAT THIS VIDEO EXPLAINED QUATERNIONS SO WELL there you go you're welcome
Explained so well💖
Thank you! I finally understand!
Thankyou for the generosity in sharing. Thankyou.
I do all my calculations with Octonians.
Excellent Video. Keep up the hard work.
That's interesting. What sorts of calculations are they?
It's funny you mentioned the octonions because I was going to make a video on the octonions and the Cayley-Dickson construction (unless you beat me to it).
It was purely in jest. I have a hard enough time time finding a practical uses for quaternions. But if I had a use for them, I would definitely use them, even if my heart is really with Clifford Algebras.
Oh damn, I was actually looking for a nice application of the octonions, other than bizarre string theory stuff.
Clifford Algebras... that is what you really want. Are you familiar with those? And if not, would a video on those be helpful? Much more intuitive and versatile.
I'm not too familiar with Clifford algebras even though I have heard the name in association with hypercomplex numbers. The topic is on my long to-do list of math topics to read about. Unfortunately, the Clifford algebra videos on TH-cam seem to be too high-level, but it would certainly be interesting if you (or someone else reading this) made a video on the topic.
Really nice explanation great job :)
thank you for the video. - good explained
Your explanation is really simple and perfect! Another question, which kind of graphics tablet You used? May I have the model? Thank you so much!
Thank you so much! This video was really usefull!! Thankss!!
thanks, really well explained!
Excellent presentation
I thought that *brief* was the wrong word to use here until I remembered you're explaining quaternions.
Hamilton is the bo rai cho of mathematics. It's crazy but it works! I juggled around proving those and damned but it worked.
I'm curious how you might go about calculating the difference between two 9 axis IMU sensors using their corresponding quaternion coordinates? I'm working on a project where I have a sensor attached to the side of the chest and another attached to the arm. My objective is to calculate the position of the arm relative to the body using the quaternion coordinates. Unfortunately, I"m afraid I don't understand them enough in order to come up with an equation on my own. Any help would be very much appreciated. Thanks!
Great explanation!
Nicely done!
Thank you so much for the video, it is very informative.
If I may have the chance to try to improve your work,
I would say to you use different colors in the writings, that add information to the image
You're the GOAT!
OMG. Your handwriting is 😍
thank you, very helpful !!
Hello,Thank you for this interesting series of videos! I just have a question: around the 11:00 mark, when multiplying both sides of the equation ijk = -1 by i , do we assume associativity applies? Just wondering this, since i(ijk) is assumed to be equal to (i^2)jk . Thanks : )
+Frank Clautier
Right, we're assuming the associative property when we do this. In a different formalism, it's provably true that quaternion multiplication is associative but I prefer setting up the quaternions in this way.
Ah ok, thank you for the clarification :)
Great video! Big shout to the Irish mathematician ☘️.
Perfect explanation:)
All lectures on quarternions are given by mathematicians, via complex numbers. Maybe the following practical application will motivate the need.
Rotations in 3D can be expressed in terms of two angles, theta and phi. One of them lies in the plane formed by two of the orthogonal axes, say X and Y; and the other in the plane involving the third axis, say X and Z. Any 3D rotation can be expressed as sine and cosine of theta and phi.
So, what is the problem? Why are the quarternions useful? Trigonometric functions such as sine are computed as infinite series. (Look up Taylor Expansion for the Sine and Cosine functions). Exact solution involves infinitely many terms. Bit real time gaming demands fast computation. Yet, truncation of a series as approximation necessarily involves errors. So what is one to do?
This is where quarternions come in; they involve only dot and cross products of real numbers - very fast and at the same time exact and precise.
Now, are you motivated to follow this or any other presentation on quarternions?
I find a good way of thinking about quaternions is to imagine to objects approaching to impact in order to cause the sum vector. This is of course a non elastic reaction, however the complex solution would be elastic and would have a loss of energy equal to the elastic energy expended. Following this train of thought this would mean that 5th dimensional dynamics would include internal variables that affect the overall motion, this would be analogous to planets colliding, or in qed with atomic emissions and interactions. But this concept seems very useful, so long as you are able to intuitively integrate the imformation.
you are the best!! thanks!!!
How is it used in the computer graphics? It always is good to listen to somebody who knows what he is talking about.
This is great. Thanks. One remark:
It's quite confusing when you point things out on your sketch and say "here, and here..." but I can't see where you are pointing:-))
At 8:30 where does ijk=-1 come from? That seems really weird because a square is multiplying two things (a thing by itself) but this is multiplying three things. In other respects I like the idea of quaternions as combinations of a vector and a scalar.
Please correct me if I am wrong, but wouldn't the answer to the (14:32) substituted chjk x-i (4 across, 3 down)be *Minus* chi when multiplied?
Thanks,
nice video and good explanation.. keep it up
0:38 Well, that's a very unusual way of drawing N
Nice job. Thanks!
very good explanation
thanks
like vectorial product.(clockwise+ counterclockwise-)
ABB industrial Robots use them - cool and thanks you overcome a barrier in understanding them for me
nice explanation... thanks..
Thank you ! very useful
Fantastic explanation. I'd like to see one with functions. Are there applications you know using vector functions and not just vectors
+Alberto Marquez
Oh sure, a lot of what we call vector calculus was originally done using quaternions. I'm pretty sure you can get the standard divergence and curl operators out of multiplying by a quaternion with the del operator in the vector part.
do you mind if i ask what your particular field of study is? is it more mathematics or physics. i was reading a book on quantum physics, and found that when the pauli matrices are considered along with the 2x2 identity matrix, it forms a quaternion. now i have only really heard about quaternions from this video. but I'm curious if you have any knowledge of this as it applies to quantum mechanics. book is called quantum mechanics written by leonard susskind
+libertyhopeful18
It's actually neither - I'm a medical student with a research focus in neuroscience. I'm merely a wannabe mathematician.
I'm extremely rusty on quantum mechanical stuff but I do know the Pauli matrices are basically a rediscovery of quaternions and (when multiplied by i) are isomorphic to the quaternions.
When you come to learn about math but end up converting to Catholicism
Pretty much.
What's the difference between 4d vector and Quaternion?
Great video
What program did you use to draw with? I have seen this program used here and there lately with game development videos and such.
+Cort Smith
I use SmoothDraw4 with a Wacom tablet.
what are you using to write?
This is the most interesting thing ever right below gravity
I'm here because I'm trying to use Quaternions for rotation in 2D Unity game dev. It's hurting my brain, but I did learn that the w component (x,y,z,w) of a unity quaternion may be a scalar. lol.
+Sparkplug1034
Yeah, it's possible that the "w" is the scalar there and they write the scalar as the fourth component (whereas I write it first). Let me know if you have any questions (comment or email me). I also have further videos on this rotation topic.
I'm trying to do a Quaternion Slerp in Unity so an enemy game object gradually rotates to turn towards the player before firing a weapon. I spent 2 days working on the trig before finding out that unity quaternions have looking at something built in :(
DUtOO Thank you. I did start using Quaternion.Euler(Vector3) , but because of what I need my objects to do, I am using other methods. Thank you though!
Are you rotating the y-axis (euler) of the enemy or z-axis (euler) of the enemy weapon? In either case, you're overcomplicating it.
crestfallenllama I'm not over complicating it in the slightest, thank you though. I'm rotating the Z axis. And it does rotate correctly. It's not supposed to just rotate though. I'm coding it's AI, using some quaternions and it's just new to me.
At 12:28, why is the 'j' placed to the left of 'i', rather than the right?
Where were you all this time? I'm so angry I just found this video out instead of years ago!
quaternions are very interesting, yet a little confusing. Awesome vid though!
I dont get why ijk=-1. Is it something I just have to accept? Assuming so then I understood the general way of multiplication of quaternions as you explained it.
same i thought ijk would have to be i^3 = -i =-j =-k
I want to offer an alternative method instead of look up table of multiplication,
suppose that
positive signed direction is i -> j -> k and negative signed direction is k -> j -> i
if you multiply consecutive two items then result is the successor(or third) item and place the sign with respect to the direction
for example,
ij is positive direction so result is +k
ji is negative direction so result is -k
ki is positive direction so result is +j
ik is negative direction so result is -j
and so on
Now do Octonions! :D
Thank you!
Thanks for your effort, it is a good work, but you didn't give any introduction about quaternions , definitions and use only the algebra. I hope you can add an introduction in future so it will be a complete session about quaternions. In general your are a good communicator.
+Hazem Demrdash
I'm not quite sure where I could have started other than just saying that quaternions are 4-vectors with a special multiplication rule. I could have gone into the history, where Hamilton was looking for a conservative extension to the complex numbers which would model 3D space, but that would have lengthened the video. Is there any specific introductory concepts about the quaternions that you think were not included in this video?
As I think about it, I recall using imaginary numbers with Mobius transforms (conformal transformations) and of course, fractals like the Mandelbrot set. Now, I have to wonder, if we can put Quaternions in place of the complex numbers in these things, do we get anything interesting?
I think I've seen some quaternionic Mandelbrot sets either on TH-cam or elsewhere online. Here's one link I found:
th-cam.com/video/AyLvyrU9SMU/w-d-xo.html
I couldn't tell you anything about the Mobius transform; I haven't worked with it.
After watching this video, there's only one question left in my mind. WHAT THE FUCK IS A QUATERNION ! ? ! ? !
+I-hate-Google
A vector with four components (w,x,y,z). It's also written as w+xi+yj+zk. You add quaternions as you normally add vectors (component by component) and multiply them following the rules for i,j,k that we derived in the video.
I appreciate the explanation as you explained it quite well in the video. HOWEVER, I'm a Unity developer (trying hard) and when we deal with rotations, we store them in Quaternions. I haven't been able to find a single visual explanation on how rotation vector components map to quaternions and why. Any ideas where I could find one?
+I-hate-Google
I'm not sure what you mean by _rotation vector_ but I've got some more detailed videos in my quaternions series showing why the quaternions work in doing rotation. Also, if you're looking for a visual intuition, I highly recommend you check out geometric algebra because in this system, quaternions are special geometric objects in 3D (not 4D) consisting of a scalar plus a bivector (an oriented patch of area). That oriented area immediately tells you in what plane stuff gets rotated whereas a 4D vector tells you nothing. In fact, I should have a couple videos discussing geometric algebra in 3D in the coming weeks.
I'll check out you other quaternion videos then. It would be awesome if you did a nice 3D visual explanation of quats (explaining how the get mapped to 4D vector). Up to this day nobody has managed that. A lot of Unity developers, including myself, would be very grateful to you. BTW there are a lot of Unity developers and I would share your TH-cam link to that video on Unity forums.
+I-hate-Google
In terms of mapping the rotation to the 4D vector (unit quaternion) if you're using the axis-angle concept of rotation, the mapping from axis-angle to unit quaternion (as you likely know) is (cos(theta/2),sin(theta/2)n_x,sin(theta/2)n_y,sin(theta/2)n_z) where n_x, n_y, and n_z are the three components of the axis (normalized). I go over why this works in the two videos that have "quaternion exponentials" in the title.
That other interpretation of quaternion as scalar plus bivector will fit in nicely with my geometric algebra series so I will cover that there.
i was trying to make a jk, but it turned out it was already equal
Also, I'm angry!
+Aiddiakapi
I guess I'm not sure what you mean in your first sentence. Good, get angry!
+mdphdguy1 jk = joke, i = jk
+Aidiakapi
Oh, I see. I didn't know "jk" was an abbreviation for "joke". Doesn't seem like it's a word that needs to be abbreviated.
mdphdguy1 It is definitely not needed, only for bad puns occasionally :).
its abbreviation of "just kidding" too
at 12:27 you multiply both sides by j. why do you get jjk=ji and not jjk=ij? How do i know where I should write that j down? Is there a specific rule?
+monatsend
Just keep left-multiplication distinct from right-multiplication.
why?
I have one question about complex numbers.
Solution of quadratic equation has:
- two or one real solutions representing intersection of parabola with X axis
- two complex solutions
Is there any special meaning of those two complex numbers?
Yes. Take the original parabola in the real numbers, let's call it z=x^2 (you'll see why I'm not using y, soon enough). Call the original parabola P1. Put the real number inputs on the x-axis, and the imaginary number inputs on the y-axis.
Make an identical copy of P1, and call it P2. Rotate P2 by 90 degrees around the z-direction vertical axis through the vertex. Now mirror P2 about the horizontal plane through the vertex.
You now have the extension of the original parabola to the domain of real and imaginary numbers. The roots of the quadratic equation that are complex, will correspond to where parabola P2 intersects the x-y plane. The x-coordinate of these intercepts is the real part of the complex root, and the y-coordinate is the imaginary part.
If we had a 4th dimension to work with, we could form the full continuous parabola of all the complex inputs in all its glory, and include the complex outputs. But, because our range is restricted to real numbers in the z-direction, it gets difficult to visualize. The parabolas P1 and P2 as I defined, are the intercept where the complex part of the solution to z=x^2 equals zero, and z is exclusively real. Many times, color shading is used for depicting the imaginary part of z, so that it ends up looking like a heat map on a 3-D surface. You may also see color shading to indicate the angle of the imaginary number, and Z-position to indicate the magnitude, where the x-y values correspond to the input to the function.