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.
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!!
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.
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 ?
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...
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.
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."
+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?
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)
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.
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.
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?
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!
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.
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.
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
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?
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.
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.
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).
@@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!
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 ^^"
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.
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
+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.
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.
+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.
+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.
+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".
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.
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
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.
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.
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?
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.
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.
I was going to point this out too. But I was betting someone else spotted the error. TY.
Yep, I was going to state that also. But Gibbs did simplified its math to vector algebra as we know today 😏
Heisenberg might have invented octonions to explain particle spins for quantum mechanics
He said Heisenberg because he wasn't certain who it was. But when he stood still he became certain it was Hamilton.
Uncertainity principle.
Takshashila underrated comment
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!!
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.
best part about watching youtube lectures is that you gain the knowledge but you don't have to do the homework
You can't gain the knowledge without doing the homework.
why not
+karz12 Word.
i prefer saying..best thing about you-tube college learning is you gain the knowledge without having to pay for it.
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 ?
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...
Hamilton, not Heisenberg.
Idiot hamilton thinking about quaternions on his way to Party
I heard Hamilton used his knowledge of Quaternions to become a drug kingpin.
I AM THE ONE WHO EXTENDS COMPLEX NUMBERS!
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.
@@random_guy6608 show some respect for your intellectual masters
So that's why electrons location are uncertain, because they're 4d beings
I watched countless videos on quaternions and this one is the best by far.
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."
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!
Excellent introduction via rotations, but the discoverer was Hamilton, not Heisenberg.
Pharap Sama
History otta' at least be in the right century...however fascinating the math...
+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?
+Lou Puls Say my name!
Yourre mothers would all b so proud
i agree too.
They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.
Aside from mistakes in mentioning History, the intuitive approach he has applied for teaching the subject, is better than many others on the UA-cam.
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)
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.
Best explanation of quaternions I've heard. Thank you.
8:36 humanity restored
You Sir rock! After too much trying, I think I understand attitude quaternions at last!
For the first time, I am beginning to understand Quaternions. Thanks, Prof!
This is an clear explanation about quarernions, thanks a lot
Much about quaternions just fell into place with your lecture, thanks.
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 😁😁
Great and impresive: Keep It Super Simple:)
Really great lecture. Thanks :D
Great presentation
Finally I get it! Very very good lecture!
Really really good lecture!!
Great! Thank you so much!
This is a valuable introduction, for people like me, who need to get started on these.
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.
Easy to understand, very good explanation.
professor i owe you for ever
His attitude of teaching shows that he is very much conscious about his topics
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?
It's interesting to know about quaternions analysis.
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!
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.
28:00 is the punch line if you're here wondering how quaternions can be used for rotations and for solving gimbal lock
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.
Quaternions were created by William Hamilton, not Heisenberg. No doubt someone has already added this in the comments.
thank you!
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.
God bles you bro❤😊😊😊
William Rowan Hamilton used for navigation gimbals, simulation motion platforms etc
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
I find quaternions applicable to statistics,also find useful the idea of (cosx+sinx.v) where v is a unit vector.
Nice lecture. I have a much better conceptual understanding of quaternions.
Does anyone have these notes that the lecturer keeps referring to? If so, could you kindly share them?
Why there is -v1.v2 when multiplying q1and q2. @17 mins
I don't understand here at 26:12, what's the theta here represents?
Where did the 5th term in quaternion multiplication come from?
Thankyou
Which website he keeps mentioning?
Quaternions were invented by Alexander Hamilton. Heisenberg was the meth kingpin on Breaking Bad. Glad I could straighten that out.
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?
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?
Quaternions start at 7:07
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.
¿it can be generalizated to a 2^n- dimentional object?, ¿ exist an n such that it forms a cunmutative field ?
Im a dumbass and I can tell that this lecture is a good one, just watch it a few times over.
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.
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).
@@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!
@@francescorizzi2601 Honestly, I just learned about quaternion rotation from Wikipedia: en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
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 ^^"
William Rowan Hamilton
The F1 Driver, not the Meth cooker
Link to notes?? Good video.
why not its also unit quaternion Im I right?
Dan WU its just the matter of which letter to chose to represent the rotation angle as a variable...
A Profundidade necessitante
William Rowan Hamilton Trinity College Dublin - discoverer of quaternions
In case someone is looking for the cheat sheet the professor is referring to:
graphics.cs.ucdavis.edu/~joy/ecs178/Transformations/Quaternions.pdf
thanks a ton I really needed it and cant access the site as I am not a student
@@pierresarrailh6617 You are welcome! So you have gotten the cheat sheet right?
@1 conscience 0 dimension Good to hear! and yea, I searched up the matrix, it seems interesting.
Hamilton?
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.
At 19:55 why is division NOT inverseDenominator*numerator in that order like matrix inverse
I don’t get the multiplication part.
...now kinda wishing I had studied computer graphics in university instead.
[breaks chalk] Well, somebody obviously supplied this classroom with right-handed chalk.
If I know linear algebra, how much time do I have to learn this
Apparently one lecture cuz he’s moving on.
4*4 matrix? Why ? Shouldn't it be 3*3 matrix?
Did he say heisenberg??? Waaaaaat
+Shoham Sen it wasn't heisenberg. The man who discovered quaternions was sir William Rowan Hamilton.
thetntm yeah i know hence the question mark... :)
That was terrible misinformation. I can only guess it is because some people are not too interested is who did what.
+comprehensiveboy
According to Simon Altmann (cf. Wikipedia) it was Carl Friedrich Gauss in 1819 (only published in 1900).
+Shoham Sen "I am the one who rotates" - Heisenberg
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
+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.
+Zeeshan Ijaz
I've come with the same conclusion, did you ever find the answer to this question?
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."
ok this makes sense now thank you.
The Sentence is : i^2=j^2=k^2=ijk=-1
😀
For q=q1×q2, how do I get q1 when I already know q and q2??
If q = q₁q₂, then
q₁ = q₁q₂q₂⁻¹ = qq₂⁻¹
Go play with 3 blue 1 brown's interactive video lectures if you want to learn about quaternions.
If one tries to define the norm of complex and others the values of i-sq and j-sq etc is equal to (- ) 1.
If i^2 = - 1, and j^2 = - 1, why should i*j not be - 1 as well?
Because then i = j and you just have standard complex numbers.
Not too sure about the Heisenberg reference.
Hamilton, not Heisenberg. •-•
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.
+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.
+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.
+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".
We want Geometric Algebra!
I don't understand these complex numbers. can someone explain how ij = k.
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.
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
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?
Spaskiba There's a channel called mathoma that has better videos on this. look for him.
Hamilton, hamilton
farewell Hamilton
nobody noticing that the inverse is missing a minus sign: students sleeping soundly.
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.
Everyone is complaining about him using the wrong name. But this isn't a history course, so does it really matter who?
Even he said Heisenberg, I am certain it was Hamilton.
Some people heard "Hamilton" and some heard "Heisenberg."
Engineers!
They get what they want.
They don't care where they come from.
The Camera Man ← → ← → ← → ← → ← → ←
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.
Hmm, wasn't Heisenberg a math cook? Lol
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?
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.