How quaternions produce 3D rotation

Wait a minute, aren't quaternions super confusing? After all, they live in 4D space!!! Let's try to put this confusion to rest.

Watch 3Blue1Brown's excellent video on quaternions:

Play with quaternions on 3Blue1Brown's and Ben Eater's interactive website:

[ebolapie]

obviously they produce rotation because they're called qua-TURN-ions duh

[swift]

I love this explanation! I watched 3blue1brown’s video first, but I still had a lot of questions. This video cleared most of those questions for me

[jamesking]

I finally understand why the double rotation happens and why you need the two sided multiplication to fix it. Thank you for this explanation!

[PrayAlways-mnwh]

Great job and this is by far the best description of a quaternion video I have come across. One comment I have is that "by definition" contains no information why things are the way they, there is no insight or intuitive feel for the description of "the way things are the way they are". "By definition" often is stated by the teacher to the student, to mean do not ask any more questions rather than giving insight. "By definition" should be explained in context, and often it means "logically consistent" or it leads to an "illogically consistent" result, in both cases the "logically consistent" and "illogically consistent" result should be explained. For quaternions, the "by definition" implies a logically consistent subfield of numbers given all pertinent rules are stated and followed. This is why Rowan Hamilton was excited and immediately scratched down the formula i^2=j^2=k^2=i*j*k=-1, he had in essence discovered a new field (to be more accurate a new subfield) of numbers, a new space, which is logically consistent given the stated quaternion rules of multiplication. The focus shouldn't be on "by definition" but the fact that Rowan Hamilton had discovered a new subfield of numbers which was only appreciated once computers and computer games became popular. To see the illogical consistency, one can attempt to create a field or subfield with i and j only, if you attempt to do this you will quickly find there is illogical consistencies within a 3-dimensional world, and one has to go to 4 dimensions (quaternions) with some extra multiplication rules to make a consistent subfield.

[oystercatcher]

This is by far the BEST description I’ve seen of quarternions brilliantly explaining both the maths and practical side of 3D rotations!

[felipeceler]

Best explanation so far. Thank you for sharing it !

[justvinchy]

Wow! I didn't imagine someone could explain is as good as you did. Great job.

[maxwellchiu]

Been looking at quaternions for a year . This is the best source so far. Much appeciated.

[goli]

I loved the video. Hamilton is one of my favorite mathematicians and I really love to talk about him. Quaternions are one of the mathematical concepts invented long time ago which have been used recently. One of the infinite many examples that answers the question "where I will be using this stuff" in math classes. Now, I can give a more clear description about how quaternions are used in animations. Thanks a lot. I sincerely appropriate it.

[Danielle-ewel]

thanks for keeping the quality so high across all your videos!

[benjaminreynolds]

I just found this channel and I am so happy that I did. Can’t wait to see more videos to come from this channel and hope it will be recognized by the rest of the mathematics community on YouTube.

[안장환-nw]

By far the BEST description after wandering all the materials.. Thanks !!

[darkexior]

Best video explaining quaternion rotation

[stevewhitt]

I have watched a lot of vids and wiki on Quaternions, but now finally I understand them. Thanks.

[noahblaine]

I am not sure who this is for. This explanation was quick and dirty and assume a very good understanding of math lingo.

[movingheadmau]

That was an amazing explanation, thank you for sharing! :)
The last part is what flashed me the most. I use quaternions for animation and rotation handling in Unity-Engine and I thought that a quaternion would only be non unique when using a rotation angle which is theta + k*360° where the sin and cos would yield the same results. I had never thought or heard of the fact that there is in fact a way to encode the "long" and the "short" way from one rotation towards another. My assumption was that quaternions would just always yield the shortest rotation. :D

[xmeansnop]

Bro, I watched a lot of stuff online for quaternions, but really nobody - and I mean nobody I've encountered - ever explain how to actually rotate things with that.
Yeah sure, there are some good explanations what quaternions really are and stuff like that, but not a single example on how to do it. Up until this moment I thought quaternionrotation happens only by multiplying a pure quaternion by a rotation quaternion and wondered why my results are so dumb.
You made it click for me and I want to thank you a lot for that.
Or how a german would say it: "Der Groschen ist gefallen."
Have a nice day.

[pavelperina]

This is pretty good. My best understanding so far was somewhat halfway between geometry algebra rotors and quaternions. Like let's rename vector space to i, j, k or e1, e2, e3. Now let's define piecewise multiplication which says that vectors i, j, k are multiplied by right-hand rule (i*j=k), but when they are multiplied by themselves, they become -1. Now use that sandwich multiplication that rotates vector by angle and adds scalar to it and then rotates it again and removes that scalar. In simple cases rotation is halfway and scalar is not produced.

By the way quaternion interpolation is NOT so easy as advertised. It's same like with complex numbers: multiplying by complex number gives you rotation. Oh, by the way ... if that complex number is on unit circle, otherwise it scales everything too. If you interpolate between two numbers on unit circle/sphere, you get closer to center and they downscale everything. Obvious solution is to normalize interpolated complex number. Hell, we have two problems now - how to normalize it when it passes through zero (interpolating rotation by 180 degrees gives us singularity just like Euler angles) and second problem is that as it gets closer to zero, angular speed increases. In the end, solution to interpolation between two orientations is to use two quaternions to find axis and angle and to interpolate that angle (SLERP - spherical linear interpolation). Then it's questionable if we want to construct quaternion or matrix, because multiplying thousands of vector by a matrix is cheaper operation.

But there are some nice uses for quaternions, such as storing orientation efficiently, finding vector and angle between two, generating random orientation, likely more complex smooth interpolation without sudden changes of angular velocity at control points.

[MGTOW-nnls]

This video is outstanding. The best explanation of this tricky material. Sir, you are low profile genius. Thank you,

[mitrafathianpour]

Thank you so much.. it was useful for me because my thesis is related to this subject and I am really interested in Quaternions because it has special role in our life.