Lecture 09: The application of Unit Quaternions to rotations
Пікірлер: 227
@lopezb8 ай бұрын
Beautiful lecture, thanks! Just the right amount of detail. Quaternions were invented by William Rowan Hamilton (also invented Hamiltonian Mechanics) in 1843. Heisenberg was one of the fathers of Quantum Mechanics in 1925.
@KaiseruSoze7 ай бұрын
I was going to point this out too. But I was betting someone else spotted the error. TY.
@joaogonzalez40825 ай бұрын
Yep, I was going to state that also. But Gibbs did simplified its math to vector algebra as we know today 😏
@gokceyildirim81614 ай бұрын
Heisenberg might have invented octonions to explain particle spins for quantum mechanics
@calmsh0t5 жыл бұрын
Praise the age of digitalization. I can get all the knowledge I want from great sources and don't need to rely on local professors who can't explain even the simplest thing, plus I can filter out the stuff that university would want me to know but I never need for what I want to do. What a time to be alive!!
@DellHell18 жыл бұрын
He said Heisenberg because he wasn't certain who it was. But when he stood still he became certain it was Hamilton.
@takshashila29955 жыл бұрын
Uncertainity principle.
@gavtriple93 жыл бұрын
Takshashila underrated comment
@JackLe11278 жыл бұрын
best part about watching youtube lectures is that you gain the knowledge but you don't have to do the homework
@karz127 жыл бұрын
You can't gain the knowledge without doing the homework.
@ZeusLT7 жыл бұрын
why not
@johnjackson97677 жыл бұрын
+karz12 Word.
@s.u.52857 жыл бұрын
i prefer saying..best thing about you-tube college learning is you gain the knowledge without having to pay for it.
@That_One_Guy...4 жыл бұрын
Advantage of online learning : 1.Gain knowledge 2. Choose to do or not to do homework (with freedom to choose when to do one) 3.Sometimes a much clearer explanation than your lecturer tried way too hard to explain (for math i loved this so much) 4. Need just a waaay shorter time time than the boring and weekly long explained things in your college 5. Free of cost 6.Never get left behind because of the bullshit limited amount time (see point 4) 7. Learning becoming much effective also because you're free from stressfull environment (annoying and noisy idiot kids who keeps babbling about something trivial, bullies) (i feel like stressful environment is one of the biggest obstacle of studying properly beside worst teaching and limited time BS) Why does offline learning isn't removed yet sigh. For anyone complaining about social interaction for same age, i ask you how does people in the past (where school isnt even exist yet) interact with each other ?
@dendrogenhs7 жыл бұрын
This lecture skips details, and the presenter does mistakes, but he really gets the intuition: this is the easiest to understand video about quaternions I ve found so far...
@realdeal9687 жыл бұрын
I watched countless videos on quaternions and this one is the best by far.
@JA-yi8bs3 жыл бұрын
A concept I was not taught at University and now faced with in my research. Your explanation has been so helpful for my understanding - thank you!
@michaell6852 жыл бұрын
Per Wikipedia, not Heisenberg (1937-1976) but Rodriguez & Hamilton in the 1840s developed Quaternions. Hamilton was its great advocate. " Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840,[1] but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations."
@APaleDot10 ай бұрын
26:40 He says the quaternion ( cosθ, sinθ v ) represents a rotation by angle θ, but it actually represents a rotation by angle 2θ. The reason: when doing a rotation, you do a "sandwich" product to prevent the vector from being pushed into 4D space, u' = q u q^-1 which applies the quaternion twice, resulting in a rotation by 2θ.
@zdspider67785 күн бұрын
Yeah, that's what I thought! It should be: _(cos(θ/2), sin(θ/2) * v)_ And he didn't explain the "sandwich" part... At least I don't think he did. And there's no "part 2".
@AlfredEssa8 жыл бұрын
Hamilton, not Heisenberg.
@random_guy66085 жыл бұрын
Idiot hamilton thinking about quaternions on his way to Party
@robrick93615 жыл бұрын
I heard Hamilton used his knowledge of Quaternions to become a drug kingpin. I AM THE ONE WHO EXTENDS COMPLEX NUMBERS!
@JimAllen-Persona5 жыл бұрын
Guess he was uncertain 😂. Another Newtonian or Gaussian type legend (Gauss’ solution to the parallel postulate). As bad as that joke is, this is my first exposure to these... very interesting.
@abenedict854 жыл бұрын
@@random_guy6608 show some respect for your intellectual masters
@That_One_Guy...4 жыл бұрын
So that's why electrons location are uncertain, because they're 4d beings
@LibrawLou9 жыл бұрын
Excellent introduction via rotations, but the discoverer was Hamilton, not Heisenberg.
@LibrawLou8 жыл бұрын
Pharap Sama History otta' at least be in the right century...however fascinating the math...
@dlwatib8 жыл бұрын
+Lou Puls He at least remembered that it was a long name beginning with H. But is it so difficult to remember that it was an Irish mathematician in the 1800s and not a German physicist in the 1900s?
@gfetco8 жыл бұрын
+Lou Puls Say my name!
@morgengabe17 жыл бұрын
Yourre mothers would all b so proud
@ahmedgaafar53696 жыл бұрын
i agree too.
@yunhyeokchoi20048 жыл бұрын
8:36 humanity restored
@englishforfunandcompetitio2482 жыл бұрын
Aside from mistakes in mentioning History, the intuitive approach he has applied for teaching the subject, is better than many others on the KZbin.
@mikedavid5071 Жыл бұрын
This is a great intuitive introduction to Quaternions. Knowing who invented quaternions gets you nowhere in understanding quaternions. Knowing the name means nothing. Knowing how to use them and forging new fields where they have practical use is quite useful.
@baruchba7503 Жыл бұрын
Best explanation of quaternions I've heard. Thank you.
@onetwoBias8 жыл бұрын
Excellent lesson! :) Impressive that he managed to make this comprehensible to someone with only a basic understanding of vector math in three dimensions, who has never heard of quaternions. (me)
@ksbalaji12873 жыл бұрын
For the first time, I am beginning to understand Quaternions. Thanks, Prof!
@slickwillie33764 жыл бұрын
They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.
@yiyangtang36229 жыл бұрын
This is an clear explanation about quarernions, thanks a lot
@shivanshiverma80253 жыл бұрын
Thank you for explaining with such an elegancy, sir! I've been stuck on this topic for a long time now, and finally you made me understand it 😁😁
@andyeverett19574 жыл бұрын
Much about quaternions just fell into place with your lecture, thanks.
@emmanuelmorales53328 жыл бұрын
You Sir rock! After too much trying, I think I understand attitude quaternions at last!
@vwcanter Жыл бұрын
This is a valuable introduction, for people like me, who need to get started on these.
@mattwolf28878 жыл бұрын
Really great lecture. Thanks :D
@benmansourmahdi9097 Жыл бұрын
professor i owe you for ever
@thejking4 жыл бұрын
Finally I get it! Very very good lecture!
@piotrlenarczyk58038 жыл бұрын
Great and impresive: Keep It Super Simple:)
@DrMerle-gw4wj11 ай бұрын
Quaternions were created by William Hamilton, not Heisenberg. No doubt someone has already added this in the comments.
@bsergean8 жыл бұрын
Great presentation
@GeForece62006 жыл бұрын
Really really good lecture!!
@SowmyanarayananP7 жыл бұрын
Great! Thank you so much!
@cyborgbeingadroidthinklike57374 жыл бұрын
His attitude of teaching shows that he is very much conscious about his topics
@miltonlai48502 жыл бұрын
Easy to understand, very good explanation.
@lawrencedoliveiro91047 жыл бұрын
Another useful feature of quaternions is that they interpolate very nicely, which is useful for animations. Say you have two orientations of an armature bone in your character. Each orientation can be represented by a quaternion. If these are keyframes, then the animation software can interpolate the intermediate orientations by interpolating the quaternions. This automatically gives you a uniform movement along the great circle connecting the two orientation points. If you were trying to interpolate Euler angles, then you would not (in general) get movement along a great circle. I think the actual curve might be a loxodrome (I’m not sure). In any case, it won’t look nice.
@JohnCena9638523 жыл бұрын
May not be perfect for some details, but definitely the best clarify of quaternion. Thank you sir. btw, does anyone know which OCW does this lecture belong to?
@NoisySoundFilms7 жыл бұрын
is there a second part of this lecture? i would like to a real application of how to move objects on 3D space. By the way! it has been a very great time seeing this lecture!
@TheSemgold2 жыл бұрын
It's interesting to know about quaternions analysis.
@pavelperina76295 жыл бұрын
34:00 please always remember original matrix, construct quaternion from original mouse position to the current one, construct quaternion (i guess there should be phi/2, but i'm not sure) and the convert it to model matrix. On mouse release store that model matrix. Otherwise they will be ugly artifacts caused by sampling of mouse coodinates and I guess rounding errors as well. PS: i have to find how to convert quaternion into 4x4 matrix, because it would be nice to visualize that in some projections. I always found q^bar * v * q as 3x3 matrix
@abhinavkumarkumar33708 жыл бұрын
Why there is -v1.v2 when multiplying q1and q2. @17 mins
@MykelGloober7 жыл бұрын
So is the V value equal to the pitch, yaw, and roll? Or is that just the vector value? Can anyone point me to a lecture that talks about vector math?
@johnhefele54323 жыл бұрын
Does anyone have these notes that the lecturer keeps referring to? If so, could you kindly share them?
@johntessin63988 жыл бұрын
William Rowan Hamilton invented ( discovered ) them. There is a wonderful neighborhood in the area called South Park in San Diego called Hamiltons that specializes in micro brews. I find a twisted satisfaction in that for some reason.
@the_nuwarrior2 жыл бұрын
¿it can be generalizated to a 2^n- dimentional object?, ¿ exist an n such that it forms a cunmutative field ?
@stevel96785 жыл бұрын
Quaternions were invented by Alexander Hamilton. Heisenberg was the meth kingpin on Breaking Bad. Glad I could straighten that out.
@liamcjbeistle32745 жыл бұрын
William Rowan Hamilton used for navigation gimbals, simulation motion platforms etc
@lunchen79852 жыл бұрын
28:00 is the punch line if you're here wondering how quaternions can be used for rotations and for solving gimbal lock
@OlivierGeorg Жыл бұрын
Good basic but approximative and incomplete explaination, which pushed me to search for more information: 1) Rotation by \phi around \vec(v) is given by q = (cos(\phi/2), sin(\phi/2) \vec(v)) 2) A position vector can be represented by p = (0, \vec(x,y,z)) 3) Rotation of p by q is given by quaternion operation p' = q * p * q^(-1). That operation is said to be computationaly cheaper than using matrices.
@TheLazyKey8 жыл бұрын
Great video on quaternions. I still don't quite understand them fully. But I'm sure applying them practically will help me fill in the gaps.
@smellybathroom2 жыл бұрын
thank you!
@phartatmisassa50359 жыл бұрын
en.wikipedia.org/wiki/Quaternion#Matrix_representations Hmmm, So I was sittin on the porch tonight thinkin, and the following is the question I came up with. Given vectors U, V elements of R3 and a quaternion (say) Q element of H s/t Q is the quaternion which rotates U to V ( as with the track-ball), Is it possible to find Q' (Q prime), i.e. the dQ/dV, or the derivative of Q with respect to the change of V s/t V rotates to U. Would that even be useful?
@KunalShah628 жыл бұрын
Where did the 5th term in quaternion multiplication come from?
@bestergester41005 жыл бұрын
I don't understand here at 26:12, what's the theta here represents?
@geoffreygoldman11156 жыл бұрын
Nice lecture. I have a much better conceptual understanding of quaternions.
@meriquirogaalbarracin24202 ай бұрын
God bles you bro❤😊😊😊
@Ybalrid7 жыл бұрын
I actually write good amonts of code using quaternions (because, 3D games and VR stuff) I never really fully understood what was these "4 numbers things", and how it can represent, well, rotation around an arbitrary axis, and why you multiply them togeter to get sucessive rotations, and all that jazz ^^"
@ogunfidodoadekunle2807 Жыл бұрын
I find quaternions applicable to statistics,also find useful the idea of (cosx+sinx.v) where v is a unit vector.
@abj12033 жыл бұрын
Which website he keeps mentioning?
@MarincasChannel8 жыл бұрын
Great lecture! But I'm still confused why quaternions actually use θ/2 instead of θ to represent an axis-angle rotation. My brain reaches a gimbal lock when thinking about this.
@BlueinRhapsody7 жыл бұрын
It is because to perform a rotation with quaternions on some 3-vector v, we take our unit quaternion p to get v' = p v p^-1. When we multiply p times v, we rotate on the unit sphere, but we also rotate into the fourth dimension [p v = (*-p dot v*, p_0 v + p x v)]. When we multiply after this by p^-1, we rotate back out of the fourth dimension by the same amount, and we also rotate forward by the same amount on the unit sphere. Basically, the first multiplication rotates us halfway there (and a little the wrong way), and the second multiplication rotates us the rest of the way there (and cancels out that 4D bit).
@francescorizzi26012 жыл бұрын
@@BlueinRhapsody please, if you can give any reference link to explain exactly this phenomenon it would be great. I'm struggling to understand this. Thank you!
@BlueinRhapsody2 жыл бұрын
@@francescorizzi2601 Honestly, I just learned about quaternion rotation from Wikipedia: en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
@micka62888 жыл бұрын
At 19:55 why is division NOT inverseDenominator*numerator in that order like matrix inverse
@jairo359 Жыл бұрын
Im a dumbass and I can tell that this lecture is a good one, just watch it a few times over.
@MatheusLB20096 жыл бұрын
The F1 Driver, not the Meth cooker
@brendawilliams80623 жыл бұрын
Thankyou
@wisdomokafor9631Ай бұрын
I don’t get the multiplication part.
@richardfantz56947 жыл бұрын
1. Maxwell's original 20 quaternions instead of the dumbed-down, truncated equations he and Heaviside later developed which is what everyone's taught in school + Nikola Tesla + Non-Herzian waves = Enough said.
@danwu72756 жыл бұрын
why not its also unit quaternion Im I right?
@user-hh7ec6bz2m4 жыл бұрын
Dan WU its just the matter of which letter to chose to represent the rotation angle as a variable...
@zdspider67785 күн бұрын
25:33 Shouldn't it be: _cos(theta/2), sin(theta/2) * v_ ? 🤔
@aylasedai23178 жыл бұрын
Hamilton?
@yb8016 жыл бұрын
4*4 matrix? Why ? Shouldn't it be 3*3 matrix?
@chanm017 жыл бұрын
...now kinda wishing I had studied computer graphics in university instead.
@Supercatzs3 жыл бұрын
Quaternions start at 7:07
@debendragurung30336 жыл бұрын
If I know linear algebra, how much time do I have to learn this
@JimAllen-Persona5 жыл бұрын
Apparently one lecture cuz he’s moving on.
@tcioaca8 жыл бұрын
Well, Heisenberg is probably the biggest inaccuracy. I would like a more solid explanation of what "gimbal lock" actually means. In this lecture, the gimbal lock is explained as if it were originating from another source of singularity inducing factor (alignment of two spatial vectors if I get his intuition correctly). A better approach to understanding gimbal lock is to _explain_ how the gimbal mechanism works. Students usually invoke gimbal lock every time their rotations work incorrectly or stumble upon a singularity.. which is not always caused by this phenomenon.
@definesigint28235 жыл бұрын
[breaks chalk] Well, somebody obviously supplied this classroom with right-handed chalk.
@zeeshanijaz28708 жыл бұрын
Around 10:00 the professor says that Heisenberg was not able to figure out ij and was forced to add another term dk to tackle the problem.Well my question is if the assumption we make is that i square = -1 and j square = -1 then it follows that ij = -1. So it is not undefined. So there was never even a problem to start with. Can somebody answer this please
@abeno628 жыл бұрын
+Zeeshan Ijaz I am no mathematician, but I don't see how you can infere that ij equals minus 1. With the assumption that i^2=j^2=-1, we only can say that i^2 = j^2 nothing more. If I follow your path, you would end up with i=j and then it's completely useless because you only get 'simple' complex numbers.
@HeliosFire9ll8 жыл бұрын
+Zeeshan Ijaz I've come with the same conclusion, did you ever find the answer to this question?
@maxwibert8 жыл бұрын
1^2=1 and (-1)^2=1, yet 1*(-1)=-1. so i have a counterexample to the argument "a^2=c and b^2=c implies a*b=c."
@HeliosFire9ll8 жыл бұрын
ok this makes sense now thank you.
@seven97666 жыл бұрын
The Sentence is : i^2=j^2=k^2=ijk=-1
@TheLeontheking5 жыл бұрын
If i^2 = - 1, and j^2 = - 1, why should i*j not be - 1 as well?
@APaleDot10 ай бұрын
Because then i = j and you just have standard complex numbers.
@ww87203 жыл бұрын
For q=q1×q2, how do I get q1 when I already know q and q2??
@edgarbonet1 Жыл бұрын
If q = q₁q₂, then q₁ = q₁q₂q₂⁻¹ = qq₂⁻¹
@SwapanChakravarthy2 жыл бұрын
If one tries to define the norm of complex and others the values of i-sq and j-sq etc is equal to (- ) 1.
@ahbushnell16 жыл бұрын
Link to notes?? Good video.
@TenHanger8 жыл бұрын
William Rowan Hamilton
@zdspider67785 күн бұрын
Hamilton invented the quaternion, not... Heisenberg. There's even a plaque on a bridge in Dublin where it "hit" him to use "ijk". He wrote it down as: _i^2 = j^2 = k^2 = ijk = −1_
@gerardoconnor42787 жыл бұрын
William Rowan Hamilton Trinity College Dublin - discoverer of quaternions
@guilhermeartigueirohenriqu20115 жыл бұрын
A Profundidade necessitante
@shohamsen89868 жыл бұрын
Did he say heisenberg??? Waaaaaat
@thetntm28 жыл бұрын
+Shoham Sen it wasn't heisenberg. The man who discovered quaternions was sir William Rowan Hamilton.
@shohamsen89868 жыл бұрын
thetntm yeah i know hence the question mark... :)
@comprehensiveboy8 жыл бұрын
That was terrible misinformation. I can only guess it is because some people are not too interested is who did what.
@ChristianS19788 жыл бұрын
+comprehensiveboy According to Simon Altmann (cf. Wikipedia) it was Carl Friedrich Gauss in 1819 (only published in 1900).
@Beon2348 жыл бұрын
+Shoham Sen "I am the one who rotates" - Heisenberg
@justbeyondthemath4559 Жыл бұрын
As a former engineering math professor I would suggest you watch his video series. He has such a good grasp of the geometrics of the math, it will make the complicated math grinding make sense when you apply it. Plus he is actually teaching this stuff not just writing stuff on the board and explaining what he is writing. BTW the very best explanation of Hamilton trying to multiply triplets and what his hang up was. The rest becomes intuitive.
@iraqplayer72705 жыл бұрын
In case someone is looking for the cheat sheet the professor is referring to: graphics.cs.ucdavis.edu/~joy/ecs178/Transformations/Quaternions.pdf
@pierresarrailh66174 жыл бұрын
thanks a ton I really needed it and cant access the site as I am not a student
@iraqplayer72704 жыл бұрын
@@pierresarrailh6617 You are welcome! So you have gotten the cheat sheet right?
@iraqplayer72704 жыл бұрын
@1 conscience 0 dimension Good to hear! and yea, I searched up the matrix, it seems interesting.
@brod5157 жыл бұрын
I don't understand these complex numbers. can someone explain how ij = k.
@henhen78907 жыл бұрын
I'm not sure if this is the right way of thinking about it, but.. I think j and k are additional dimensions in the imaginary space, much like how we have x y and z. Where if you cross x and y you get z, they complement each other.
@alfonshomac7 жыл бұрын
So remember that i^2 = j^2 = k^2 = ijk = -1 get ijk = -1 Multiply both sides by k, consider each of the following lines as steps. ijkk = -1k ijk^2 = -k ij(-1) = -k because k^2 = -1 -ij = -k ij = k
@dangiscongrataway23657 жыл бұрын
how does this work exactly? ikj=-1 but i^2=k^2=j^2=-1 that doesn't make sense It doesn't make sense to me, why i=k=j isn't true?
@alfonshomac7 жыл бұрын
Spaskiba There's a channel called mathoma that has better videos on this. look for him.
@pianochannel1002 жыл бұрын
Go play with 3 blue 1 brown's interactive video lectures if you want to learn about quaternions.
@justbeyondthemath4559 Жыл бұрын
Quaternions are the first step to fixing Euclidean space. To the beginner, you can think of i,j,k as 90 degree rotations in the respective planes. Just like the Argand plane (complex plane) or i plane in the quaternions. 1xi = i ixi =-1 -1 x i = -i and -i x i = 1 which puts us back to where we started. BTW I right multiplied to show you next state but technically it should be left side.
@8cccpeevostokzempf9 ай бұрын
Not too sure about the Heisenberg reference.
@TheRCrispim7 жыл бұрын
Hamilton, not Heisenberg. •-•
@joehsiao62248 жыл бұрын
Question: The professor said we can multiply a series of quaternions and convert the final result to a 4x4 matrix and use it with other transformations. But can't we just multiply the 4x4 matrix of each rotation and get a final 4x4 matrix that way? Isn't this how cumulative matrices work in OpenGL?
@lawrencedoliveiro91047 жыл бұрын
I think the main problem is not the computational expense, it’s the potential for rounding errors. Say you are doing an animation, with repeated accumulation of lots of small rotations. The resulting matrix ends up no longer representing a pure rotation, it adds some distortion to the object as well. Using a data structure that can only representation rotations (like quaternions) avoids this problem. You accumulate the quaternion, then convert to a matrix at each step. At the next step, you don’t use that matrix again, you compute a new one from the updated quaternion.
@AndreaCalaon733 жыл бұрын
We want Geometric Algebra!
@xusv-hi4kl2 жыл бұрын
😀
@streamapp9 жыл бұрын
I believe this is a "dot product inverse", not a multiplicative inverse. Loosely saying inverse is a rather dangerous things when teaching students because they might not realize that you can't actually multiply two quaternions together due to dimensional mismatch.
@dlwatib8 жыл бұрын
+David Jackson Except that you *can* multiply two quaternions together. That's kinda the whole point! View the video again. You use both the dot product and also the cross product when multiplying.
@techeadache8 жыл бұрын
+dlwatib +David Jackson After all that work, 19:20. The inverse (reciprocal) is wrong. V should be negative V. The reciprocal of a quaternion(q) is its conjugate(q*) divided by its norm squared(||q||^2). The conjugate(q*) is written as q* = (a, -V). The norm is referred to as the length in this video. So it turns out that everyone is wrong. Rejoice! Except dlwatib. He is still right. This professor has trouble with negative signs. But it is the concept that matters. Too bad no one understands the concept. 150 years later, Sir Hamilton is still owning us.
@Math_oma8 жыл бұрын
+d jax What is a "dot product inverse"? It is a fact that all nonzero quaternions, that is, the quaternion (0,0,0,0), have an inverse. Furthermore, any two quaternions can be multiplied together, there is never "dimensional mismatch".
@Pengochan8 жыл бұрын
nobody noticing that the inverse is missing a minus sign: students sleeping soundly.
@McTofuwuerfel6 жыл бұрын
Even he said Heisenberg, I am certain it was Hamilton.
@williamolenchenko57723 жыл бұрын
Some people heard "Hamilton" and some heard "Heisenberg."
@diabolicallink6 жыл бұрын
Everyone is complaining about him using the wrong name. But this isn't a history course, so does it really matter who?
@housamkak6465 жыл бұрын
farewell Hamilton
@rajdipde30585 жыл бұрын
Hamilton, hamilton
@rasitcakir96803 жыл бұрын
Engineers! They get what they want. They don't care where they come from.