3D Rotations in General: Rodrigues Rotation Formula and Quaternion Exponentials

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.

[@pierret00]

I am blown away by your series on the derivation of Rodriguez formula and its quaternion equivalent! The fact that you're willing to share your gift of teaching intricate maths concepts with the world for free is just wonderful. Thanks so much and keep up these videos!

[@iwonakozlowska6134]

This is my favorite video about quaternions.

[@j50313j50313]

Most comprehensive and thorough guide to 3D rotation! Thanks!

[@ewanfawns547]

Thank you so much. I've been looking for over 8 hours from reading several books and looking at several videos for someone who isn't a complete weirdo to explain this to me like a normal person. You are actually the only one.

[@robinhilliard]

This series has untied decades of knots in my understanding of quaternions - bravo!

[@andranik3949]

This video is sooooo valuable for someone (definitely not me) who's been trying to get a grasp of quaternions magic formula for the last week! Thanks a lot!

[@thegozer100]

This is so awesome! I've been trying to understand quaternions for quite some time but I was never able to find a good explaination. This video made it clear after watching it one time and without pain

[@nandha0150]

Thank you very much. Best quaternion explanation and rotation video ever in KZbin. Love from India!

[@danielssandu5578]

Very clear, it is exactly what I expected to see and understand fast. Thank you!

[@sibonelomabuyakhulu2329]

You earned my subscription. I have been trying to understand the derivation of the Rodrigues Rotation formula for a long time now. Many thanks.

[@siddgangadhar1234]

I think, of all the videos I found on youtube regarding quaternions, I found your video playlist most informative and meaningful. It's amazing how you've so effortlessly explained a rather complex (wink wink!) topic. Thank you so much for you time and effort.

[@JoseML564]

Dude, seriously thank you very much for create this videos. Your explanations are awesome.

[@AlMQTB]

Immensely thank you, author! Great detailed explanation!

[@williejohnson5172]

THANK YOU!!!!! I've read several books trying to understand how this quaternion rotator was derived, that is q[p]*q. All the books have the quaternions making these inscrutable loop de loops. You cleared that up for me quite nicely. I don't know if you know it or not but the rotator is actually a derivative. Remember your v= vparallel + vperp equals the hypotenuse which is equal to the tangent which is a derivative. The rotator rotates the tangent to a new position v'. Also if v is perpendicular to n then (v'/v)=e^((theta/2)n)=y/x=rate =f'(x). Note the theta/2 is there because of Euler's formula. e^((n(pi/2))i ) where n is now some integer multiple of pi/2 radians. I've reawakened Hamilton's and Tait's use of the quaternion for differential calculus.

[@scientifically5812]

Too bad that I have not found you 7 years ago. Great work.

[@ju-seungbyun1203]

Wonderful!! Best explanation, I've ever seen. Thank you very much.

[@Math_oma]

+juseung Byun You're welcome

[@jmsequeira]

Thanks to you, I've got it. Please, keep publishing those videos. Thanks a lot! :-)

[@astronautjupiter4225]

Finally I can implement asynchronous timewarp for my VR project thank to awesome Mathoma!

[@rahularvindshinde]

This series helped a lot. Thanks!

[@mefuri_k]

The book i use told me to create a rotation matrix to rotate points (interpreted as vectors) which is programatically slow. Yep, then instead of using rotation matrix, I'll just operate it straight to the points, now it runs around 100 times faster. Thanks mathoma.

[@fatimamirza9510]

THANK YOU SO MUCH :D. SO HELPFUL. MAY GOD BLESS YOU!

[@neelarjuna2536]

Thanks for the video dude. Keep uploading still more.....!

[@jonathanpearl9180]

After watching through all the videos in this series, getting to 28:08 felt like a christmas present all on its own. As an engineer who deals a lot with 3D orientations, quaternions pop up all the time. It's nice to finally understand where all of the "black magic" comes from rather than the dull unexplained equation slapping that goes on in other videos of the same subject. Happy Holidays, Mathoma -- I'll definitely be sharing these videos with my colleagues.

[@Math_oma]

+Jonathan Pearl Then 28:08 is my Christmas gift to you engineers. Merry Christmas!

[@kenfaulds8818]

@@Math_oma thanks very much appreciated.

[@osamahasan3288]

excellent presentation, as usual. Thanks.

[@ian.ambrose]

Thank you, such a beautiful professor you are.

[@pratapbhanusolanki6613]

Hey @Mathoma, these videos are clear and to the point. Well explained with good presentation too. I am wondering if you can or already explained a physical (geometrical) significance of a quaternion somewhere?

[@user-qk1si5cd9m]

It was a fantastic tour... Thank you.

[@yxhuang3239]

Thank you so much for these videos!!

[@PauloSustelo]

This is the most clear explanation I saw on the subject quaternions. Thank you very much. After seeing your explanation of the compact formula with the exponentials, I wondered if it would be possible to use the properties of exponentials to make successive rotations. But I have blocked at this point : the multiplication of quaternions does not comute, however the addition is clearly commutative. Why adding the exponents does not work?

[@NGBigfield]

27:45 I just sat alone in my chair, and clapped. That was brilliant !

[@SaadTaameOfficial]

Mystery of rotations with quaternions resolved. These videos about rotation are going to help me implement a Quaternion object to be used in my game projects. Far better explanations than any other videos I have seen or any articles I have read about rotations with quaternions. Keep up and thank you :)

[@Math_oma]

+Saad Taame Thanks! Not to be too cocky, but most of the other videos on KZbin about quaternions are quite bad, poorly thought out, and do not explain why the formulas work or where they came from. As you can see, the rotation formula isn't conceptually much different than working with complex numbers in 2D and it isn't difficult to explain if you tackle it step-by-step. A lot of other references just present you with the formula with no other background and that leads straight to confusion.

[@SaadTaameOfficial]

That is true and there is a reason for that: almost all of the people who cover these things come from a computer graphics background and they've learned the formulae in the same way they present them.

[@Math_oma]

+Saad Taame For me, I have zero computer graphics experience and much more physics and math experience. My particular interest is that quaternions are a special case of a more general topic called "Clifford algebra", so that's why my approach is almost totally mathematical.

[@SaadTaameOfficial]

I have been watching your videos a lot lately. I really like how you explain things. I am a programmer and curious about maths. I find it really depressing when the subject matters but the person explaining it skips important details. But you are doing a great job really, keep it up ! I wrote a blog post about rotation using quaternions if you want to check it. Link: blog.saadtaame.org/2016/09/matrix-representation-of-quaternion.html

[@Math_oma]

+Saad Taame Also, I forgot to mention that you might be interested in geometric algebra, which is a generalization of complex numbers, quaternions, and vectors. I'll probably make some videos on this topic eventually. There are some subtle conceptual difficulties with using quaternions as vectors because they are actually pseudovectors.

[@Sopiro]

Thank you so much! Amazing explanation!

[@spicemasterii6775]

Wow! What's this? Magic?! Amazing video. Thanks for clear explanation.

[@Math_oma]

It is quite miraculous that quaternions do this rotational job pretty well, though admittedly not as well as Clifford algebra.

[@ScienceDiscoverer]

The Essencial math for game programmers book just skipped thorou all of this steaps, presenting quaternions like magic. Now, thanks to you, I know its logic, not magic!

[@waqarhussain2510]

I can just say a great job because you removed the vagueness.

[@ian.ambrose]

12:44 Hi, do you know what projection formula this is? Where can I read the derivation of that formula? Thank you!

[@debendragurung3033]

Sweet, please keep them coming

[@roberthayter157]

Really wonderful video. Thanks.

[@rosyluo7710]

Very clear explanation! THX a lot ~

[@cgcode]

Thank you for awesome^10000000 video. It helps a lot!

[@vishalramadoss668]

Very well explained and presented

[@vivekdabholkar5965]

Very nice indeed! Great videos and friendly explanations. Are there any other such videos?

[@leiladousti3769]

excellent, Thank for your explanations .

[@Ehs_Houssam]

Such a good explanation! thnx

[@314dez]

Thank you Sir! Two more hystorical questions, if you allow. 1) Why are fairly simple cross and dot products (which formula I discovered for myself by simply multiplying and summing up other components: (v1y*v2z+v1z*v2y, and etc.)), were researched after quaternions, which are more complicated? 2) How could Rodrigues use a cross and a dot products if his (Rodrigues) formula was researched before quaternions?

[@arthuran4361]

good videos, easy to understand...

[@mustafasabri6430]

Wow, that was really a great interpretation for quaternion, can you please share to us the reference that you used for this video. I really appreciate your great effort

[@darovan4398]

You are a legend !

[@abnereliberganzahernandez6337]

nice videos I needed to know the derivatives of quaternions

[@nizarouarti1312]

I wondered if you are writing books about geometry, because you are very good at it. I am always amazed by the quality of your explanations!

[@Math_oma]

No, I've just studied and thought about it a while.

[@pixelrobotics6399]

@@Math_oma Because of people like you, I think internet can be a beautiful place. Thank you very much for the time you spent to share your passion! It is very communicative!

[@arbab64]

Great lecture

[@krelly90277]

Outstanding.

[@MrGeorgerififi]

AT LAST i get why rotation needs the inverse in v' = qvq^-1 !!!! thank u so much

[@cpsctutor168]

Thank you so much. Save my project night

[@Math_oma]

+wei wei You're welcome.

[@ilafya]

You are the Manˋ

[@shaypatrickcormac2765]

Better than video of 3blue1brown guy

[@u-lo-l]

TRUE NEWS

[@krelly90277]

3brown1yellow

[@maxwellsequation4887]

Nice joke

[@ilafya]

You are the Man

[@matematica_do_zero]

Saved my life! Thanks!!!

[@HearTruth]

thank you for this ...

[@bernhardriemann3821]

the video was just amazing, it is probably one of the best math videos on youtube. just one question, does this formula work for rotating points in higher dimension beyond 3d.rodrigues formula utilizes cross product , but cross product dont work on dimensions higher than 3 except 7, so are there other formulas for rotation in higher dimension or is it impossible to rotate vectors in higher dimension?

[@npathegenius5733]

These formulas do not work in dimensions greater than 3 other than 7, because there is ambiguity regarding the orthogonal axis’ to the axis of rotation. For example, in 4D for any axis of rotation there are more than 1 possible planes of rotation. If you wish to extend rotations to n-dimensions, consider using defined planes of rotation. These are used in the form of Bivectors in Geometric Algebra, which mathoma has an outstanding series on.

[@bernhardriemann3821]

@@npathegenius5733 I know there are multiple planes of rotation in 4d by some linear algebra but are there any formulas for rotation if a particular plane of rotation is defined?

[@npathegenius5733]

@@bernhardriemann3821 Yes, though the only ones coming to mind are from Geometric Algebra. e^(theta/2 * PoR) * v * e^(-theta/2 * PoR) Where PoR is a bivector and v is a vector.

[@bernhardriemann3821]

@@npathegenius5733 thanks

[@bouchemalbillel3285]

thank you very much

[@andrerossa8553]

thanks a lot!

[@ian.ambrose]

12:44 Hi, do you know what projection formula this is? Where can I read the derivation of that formula? Thank you!

[@b43xoit]

The final form of the rotation formula looks like some of the matrix formulae that come up in quantum mechanics, but in one case you see the quaternion conjugate and in the other, the Hermitian conjugate of a matrix of complex numbers.

[@Math_oma]

+B. Xoit Yes, there are many similarities to physics, that two-sided operator being one of them. Another connection is that the Pauli matrices are basically a rediscovery of the quaternions.

[@rpgtalkout8793]

Beginner trying to understand the initial equation, why does "^n" or the length of the angle of rotation factor in at all?

[@smotala11]

At the start what function is the plane and therefore n performing. Surely the plane can always be defined so that v is on it?

[@sheidadanesh]

AAAhahahah "that mushroom asshole!" really helpful and clear video. Thanks!

[@Timo4eus]

This is gold, thanks.

[@ilafya]

Thank you twice

[@066490701]

Hey guys, where exists clear explanation of quaternion exponent form from this person? P.s. great video with simple explanation of complex things, respect to the author!

[@rvoros]

kzbin.info/www/bejne/bmmlcmuXhJikjqM

[@lucasvargas7515]

i'll use for simulation in racing cars

[@jamesbean9396]

Does anyone know where the original video that goes over the derivation of the Rodriguez equitation is? as far as i can see the previous video in this series doesn't cover it? Apologies if i've missed it! Great video regardless :)

[@vengeance6051]

Search on his playlist, he has only two videos on 3D rotation

[@carlosrojas5125]

Thanks

[@username1125]

Can you rotate a vector around an axis which it doesnt intersect or connect with?

[@moseschuka7572]

I would give a super like if I could. Thanks

[@johannesugb]

Man, this is so cool. Thank you so much! Are you still making math videos? What's going on with your patreon? Is that still active?

[@johannesugb]

Anyways, you've got yourself a new patron. I hope that we'll get new math videos in the future. Your explanations are extraordinary.

[@ian.ambrose]

12:44 Hi, do you know what projection formula this is? Where can I read the derivation of that formula? Thank you!

[@SimchaWaldman]

30:58 Well said! #VacuousTruth

[@gigi12gigi12]

Hello! I watched all of your videos on the quarternions and understood most of it, but I still have unanswered questions. 1. When you refer to turning your v vector to a quarternion at the end of the video(30:00), do you mean it as a 4D vector with 0 as a scalar component, or a matrix as you showed it in the "Quaternions as 4x4 Matrices" video(7:15)? 2. If you use the cos-sin formula, isn't it possible that you will end up with a 4D vector with not 0 as a scalar =>you can't turn it into a 3D vector again? Probably I would have the answers to these questions if you had showed an example with the formula at the and of the video, but you hadn't. Maybe could you make a short video with some examples about using the formula to rotate vectors?

[@Math_oma]

+gigi12gigi12 Yeah, when I say turn the vector into a quaternion, I mean let the scalar be 0. And when you use the rotation formula, you'll always get a quaternion with a 0 in the scalar part. It's a little tedious to show that, but it can be done. As far as examples, I'll consider making a little video on it, but it's something that is quite common on KZbin. This formula is a little tedious and you'd probably want to write a program to do the calculation for you. Usually people do the opposite - show all examples but never go over why the formula works.

[@gigi12gigi12]

Thabk you for the quick answer! I just wanted to ensure that I understood everything correctly so I won't make errors in my code but I'll probably handle the coding and do some tests afterwards. Anyways, thank you for the detailed explanations! They helped me very much!

[@Math_oma]

+gigi12gigi12 Well I hope this stuff works. There's another way to think of rotations using "geometric algebra" which is more natural and is completely in 3D (no 4D quaternions). You get almost the same formula, but quaternions will do the job.

[@gigi12gigi12]

Just a quick note: I managed to solve an example and I indeed got 0 for scalar so I calmed down :)

[@Math_oma]

+gigi12gigi12 Haha, very good! Like I said, if you slog through the computations you could prove that you'll always have a 0 in the scalar part if the thing you put in has a 0 in the scalar part.

[@lt4376]

7:15 Would the V parallel formula term be zero if one had the plane that the V prime and V made together? In other words, if one had information on the axis of rotation that V and V prime rotated about, not just the perpendicular components of V and V prime, then the V parallel formula term would be zero - 14:40. Will you give some situations where one would rather use an axis of rotation in which the perpendicular components of V and V prime rotated about than using the axis of rotation for V and Vprime? Thank you.

[@Math_oma]

+Louis Tkach I don't quite grasp your question - perhaps you could try reformulating it. Does the video immediately before this one answer any of your questions?

[@lt4376]

Doh! I thought that the axis of rotation was different for this video (which had vector components parallel to the axis of rotation) than the axis of rotation in the previous video. So, the axis of rotation remains the same (in this video as the one in the previous video) despite having parallel vector components along the axis of rotation - all other things being equal. Is that true? Thank you.

[@Math_oma]

+Louis Tkach I'm considering any arbitrary axis of rotation in both videos. In the _special case_ video, I only consider rotating vectors completely in the plane defined by the axis (equivalently, orthogonal to the axis). In this video, I consider rotation of any vector, which will, in general, have a component parallel to the axis and a component orthogonal to the axis.

[@lt4376]

Alright - thank you

[@mollejalopez8012]

❤

[@yogeshjainbhansali8082]

Nothing but thanks!

[@Vastasoceans7532]

Which is the previous video?

[@MotraZotra]

I am still confused what Quaternions ARE. A quaternion with a scalar 0 can be thought of as a vector? But a quaternion with a scalar cos(theta) can be thought of as a rotation? I guess what would complete the missing link for me is when you can think of quaternions as vectors and when as rotations, when both or something else? For instance in complex numbers if you put the real component on the x axis and the imaginary component on the y axis. You can imagine i as the axis y. Multiplying i by i can be thought of as rotating i with the already rotated i. Which rotates it into -1. So a vector could be thought as both a position in the system and as a rotation of the 1 vector into it's own position? I am not sure how to transfer what seem intuitive in complex numbers to quaternions.

[@SandburgNounouRs]

Could you make the transposition to computer quaternions [4x4] matrices at some steps?

[@Math_oma]

+SandburgNounouRs Sure, you can always convert the ordered 4-tuple form to the matrix form using the matrix form shown in the other video. Also, if you need a conjugate of a quaternion in matrix form, that would be the matrix transpose.

[@may8049]

Legend

[@ian.ambrose]

12:44 Hi, do you know what projection formula this is? Where can I read the derivation of that formula? Thank you!

[@may8049]

@@ian.ambrose it's the dot product projection, kzbin.info/www/bejne/gqqqfKyZjrllrJI

[@ian.ambrose]

@@may8049 Thank you. Really appreciate your help.

[@kablouserful]

Holy balls!

[@ian.ambrose]

12:44 Hi, do you know what projection formula this is? Where can I read the derivation of that formula? Thank you!

[@seculardharma]

I have a feeling the equation does not take into account only the rotation parallel to the plane. But what do we want if we want a general rotation in any axis.

[@Math_oma]

+anmol monga Do you have a counterexample?

[@seculardharma]

I mean by , If we rotate the the vector parallel to the plane only then the same rotation will be projected onto the plane. If rotation occurs in direction with components in the line perpendicular to the plane . The same angle will not be projected onto the plane.

[@Math_oma]

+anmol monga I'm still not sure what you're thinking of. Can you give an example (counterexample) where the formula doesn't produce the correct rotation? I divided this topic into two parts to handle the special case where the vector is orthogonal to the axis (equivalently, in the plane of rotation) then the general case.

[@AnikHalder]

Wonderful video! Thanks a lot! I think anmol monga wants to say that for the general rotation you've depicted in this video, the vector v is rotated through an (azimuthal) angle theta while keeping the polar angle (angle between v and n) fixed. I presume the question is what happens when the polar angle also changes? I guess we then need to treat the azimuthal and the polar rotations separately.

[@freshofftheufo]

love your lessons, I don't even have to smoke dope to understand them!

[@pratiktohidayat1746]

Sir, I don't understand why "e^n = cos(theta) + sin(theta) nx i + sin(theta) ny j + sin(theta) nz k" and not "e^n = cos(theta) + sin(theta) nx i + cos(theta)+ sin(theta) ny j + cos(theta) + sin(theta) nz k"? Thank you

[@pratiktohidayat1746]

in 18:21

[@Math_oma]

+Pratikto Hidayat First, remember that's e^(theta*n) not just e^n. I cover that in a little more detail in the previous "special case" video, but imagine you had e^(theta*i) instead of e^(theta*n). By Euler's formula, we know that's e^(theta*i)=cos(theta)+sin(theta)*i. Now, just replace that "i" with an "n" to get e^(theta*n)=cos(theta)+sin(theta)*n. Remember that n is a unit vector n = nx i + ny j + nz k which squares to -1 just like "i" does, so replace that into the previous equation to get e^(theta*n)=cos(theta)+sin(theta)*(nx i + ny j + nz k) and distribute that sin(theta) to those three terms to get the final quaternion e^(theta*n). The key conceptual thing here, I think, is that n squares to -1 (if n is a unit vector) which means you can manipulate it in these exponentials just like you might be used to with complex numbers.

[@pratiktohidayat1746]

okay, I get it. thanks for explanation

[@hariharanramamurthy9946]

sir, is v_vec.sin(theta) = |v_vec|.sin(theta)

[@mathematicsclub6740]

Hello how come that any q= (cos@,nsin@) and how do you derive e^n@=cos @ + nsin @ so cos @+ isin@+jsin@+ksin@ Where n is the base of 3D so it is i , j , k does that come from the fact that n = i + j +k and Taylor has thereby actually 3 Infinite sums corresponding to sinx . Thx

[@mathematicsclub6740]

And also why it is somewhere @/2 and here just @

[@Math_oma]

+mathmaticsclub I've got a separate video on Euler's formula for quaternions if you wanted to check that out. When the quaternion is q=(cos(θ),sin(θ)n) where n is a unit vector, this means n squares to -1 under the quaternion product - this is exactly what an imaginary unit, i, would do in the complex numbers so I can equate (cos(θ),sin(θ)n)=e^(θn). Notice how you're just replacing i in Euler's formula with n. Furthermore, expanding out (cos(θ),sin(θ)n) gives me: cos(θ)+n_x*sin(θ)i+n_y*sin(θ)j+n_z*sin(θ)k where n_x, n_y, n_z are the components of the unit vector, n. As to why formulas sometimes contain θ/2 and sometimes θ comes from the derivation I have at the end of this video, so I'd recommend reviewing this. In general, rotation acts as a two-sided operation with half-angles in the arguments and in special cases simplifies to a one-sided operation with the full angle in the argument. The half-angle also arises from the fact that double reflections produce an equivalent rotation, but a rotation by twice the angle between the vectors responsible for the double reflection. That's really the reason you have to actually start with half the angle of the desired rotation in the general two-sided rotation formula. On that point, I'd recommend checking out the geometric algebra series to learn more about that, as it's much clearer in geometric algebra compared to quaternions.

[@MagneBugten]

Omg, i've spent hours listning to quaternions now and im none the wiser. It's not that you don't explain well, because you do, but this is all purely mathematical. I have: q0 = -0.329936 q1 = -0.045088 q2 = 0.114060 q3 = 0.936001 And i still have no idea what to do with them! I mean yes im drunk, and its way to late but I must have fallen off at a very early stage here. I guess I can theoretically use quaternions to rotate vectors in three dimensions, thanks to you, but damnit! I'm sure this is just one of those things mathematicians invented to piss off engineers for fun. Please help.

[@Math_oma]

+Magne Bugten Actually just by examining your q1,q2,q3 I read off the axis as (-0.045088,0.114060,0.936001). No need to normalize it like I did above. That scalar part q0 tells you by what angle you're rotating along this axis, remember that's q0 = cos(theta/2).

[@Math_oma]

+Magne BugtenActually, if you've understood theoretically what quaternions are doing, I've done my job. Many people have trouble with that but can crunch the numbers. But computers are for crunching numbers, right? We humans do the theory.

[@MEan0207]

Why theta/2

[@mrigank8822]

Quarternions FTW. Euler basically fixed maths by fucking over sin and cos.

[@nickelodeonserije5130]

H

[@abublahinocuckbloho4539]

you have a habit of misspeaking. at about 8:01 you say "and v parallel is going to be eventually rotated", but you had just set up the argument that v perpendicular is going to be the only component of v that rotates. you might want to make sure you dont contradict yourself or misspeak as it can be confusing to follow on from that point.

[@ian.ambrose]

12:44 Hi, do you know what projection formula this is? Where can I read the derivation of that formula? Thank you!

[@fefeisbored1958]

Why the hell do you need Quarternions for this?

[@derekxiaoEvanescentBliss]

quaternions resolve the issue of gimbal lock.

[@Dan474834]

It's too complicated, which is why they eventually just switched to using rotation matrices

[@Math_oma]

+gfgf fgff What's too complicated?

[@Dan474834]

The application of quaternions to 3D rotation. That's why their use is obsolete in most problems.

[@Math_oma]

+gfgf fgff That might be true, although I've been told by many people that quaternions are the superior approach. At the theoretical level, quaternions and related concepts are the most natural objects for describing rotations - the simplicity of the formula is a good clue that this is true.

[@Dan474834]

Perhaps the concept may be more intuitive, but using algebra is so much easier in practice.

[@0LoneTech]

Quaternions are far from obsolete. They're heavily used where we want to separate the operations of translation and rotation. A 4x4 matrix can express rotation, scaling, translation, shearing and reflections, as well as be applied to another of its kind. When we want only rotation and translation, it can be reduced to 3x4, but is still a bit hard to verify that it isn't reflecting, reducing to a plane, etc. If we use a vector and quaternion, we can verify the quaternion is of unit length far easier, and it takes 7 scalars rather than 12 (or 16). In storage, transfer and even calculation it is common that size is more important than number of operations, so e.g. a skeletal system based around quaternions has large advantages, even if the final rendering form is through matrices. Another example I've encountered recently is a 6DoF tracking system, where the rotational frame may not necessarily be aligned to the translational frame yet.

[@eprstsrpe]

Hmmm... Rodriguez invented his formula years before Hamilton invented the quaternions. But the dot product and cross product used in the Rodriguez formula came from quaternions. I'm confused 🤔🤔🤔