Quaternions and 3d rotation, explained interactively

amazing .. the interactive site is beyond cool. use quaternions in 3D fractals, where the 4D structure is most useful and interpolation methods are important. edit: did you and ben write the java for that site? the experience is entirely surreal, sortof like when the physics lectures suddenly make sense once the LSD kicks in

Please explain Dual Quaternions in this way would be amazing to see

What software are you using to do the 3d rotations?

After trying to program 3D rotation and struggling with gimbal lock while not knowing what they is, I'm glad to finally hear of quaternions.

I JUST REALIZED SOMETHING.... every time you have something where the total size is 1 it's a normal vector.

and I've heard that dot product of two vectors is essentially a cosine of their angle, SO does this mean that we can go from normal vectors being described as euler angles, into a quaternion rotation?? like it feels like it should be possible.

puts on tinfoil hat everything is just circles and triangles

Ironic that something mechanical would get locked up at its joints even when you Euler down...

A line rotates around a point, the latter being element of the line. this rotation describes a plane.
0D 1D(rotating) 2D

1 dimension higher:
a plane rotates around a line, the latter being part of the plane. this describes a space.
1D 2D(rotating) 3D

next level (it cannot be imagined, but is a mere analogy of the former 2 rotations):
a space, eg a sphere, rotates around a disc, the latter being element of the sphere, it is the disc inside the equator. this rotation describes a hypersphere (does it?).
2D 3D(rotating) 4D
Can this be calculated and shown with quaternions?

would love a response to get this straight:
-rotation is 3d, and can only rotate in 3 dof so eulor or any crude way to move from 1 point to another in 3d is to zig zag, quaternions are a linear uni-approach
-you can't actually rotate along an axis that intersects between the 3 planes, but qauternions allow you to do this.
-people get confused because of local and global space, and that the way they work is to a sphere moves independently from global space and then locally rotates (bit of a bda explanation but you get the jist).
-hyperbolically, ther is no rotation, just stretching, but the equation perfectlly stretching in one direction has diametric "in-stretching" to give the illusion of rotation.
-people are thinking of vectors and not about the whole sphere

eulor angles also seem to be terrible for coding. sequenced rotaiton can make interpretation and accuracy terrible

what is the app thx

Here’s a small experiment to see how Gimbal Lock works:

Place your phone face up on a surface, with the charging port facing you. We’ll define the phone’s center as the origin, right of the phone to be positive x, up from the phone as positive y, and away from the phone (the side with no charging point) as positive z.

When I say “rotate clockwise” around an axis, that will mean facing the positive axis from the origin and rotating clockwise from that perspective.

First, rotate your phone 90 degrees cw around the y-axis. Your phone’s charging port should now be facing to the right.

Now rotate your phone 180 degrees around the z-axis. Your phone should now be upside down with the port still facing to the left.

Alright, keeping that in mind, we’ll start again.

This time, your first step will instead be to rotate your phone 180 degrees around the z-axis. Your phone should be face-down and have it’s charging port facing you.

Next, rotate your phone 90 degrees around the y-axis. Your phone will now be face down with its port facing to the right.

Notice how in both scenarios, we used the same two steps, only in a different order. However, in the first situation, the charging point was to the left while it’s to the right in the second one.

This phenomenon is known as Gimbal Lock, and is significant because this type of rotation is how 3D rotation matrices work.

Quarternons are necessary here to keep these steps consistent.

Hi, thank you for the video.
How can I rotate around another point than the origin?

I've found this video by accident, and I've found your site with interactive demos through it. I think I finally got it. Because quaternion rotates around two 4-dimensional circles, and you only need rotation of one, you use two of them to cancel rotation of the second circle.

thanks for that, i have understood the concept very clearly

A stupid question from some old guy who is "out of business" (never worked in the field after study almost 50 years ago): why are the four dimensions not all interconnected with the cross product, something like
e x i = j ?
Why is the "real axis" direction not included in the cross product family?

this is like crack cocaine for nerds lol

I do not want to imagine rotating something in 4D 💀

By 'bugs and edge cases' you mean that everything goes FUBAR at angle 0 or position 0 because regular mathematicians wont let us have -0. ie, facing to origin has to be stored separately to current rotation and position. which is basically 2 to 3 times the computations. From what I understand Nvidia has basically gone ahead without the maths communities permission and implemented -0 at chipset ;)
ALSO: your interactive videos are fantastic! I have a passion for CAL, I wrote my thesis on it. Good job Team!

the header had me actually dying laughing

Sorry, you are talking much to fast !!!!!!!!!!!!

what

If i understand correctly then the sphere we end up with is a tangent of the 4D hypershpere at a certain location. Traversing the hypersphere results in different tangents and therefore different spheres. Multiplying this with the inverse of the hypersphere again ensures that certain properties (volume) are maintained. From this I have two questions that might help my intuition.

1) A tangent line touches a circle at one point, both for the line and the circle, how does this extend to the tangent of the hypersphere? Is there a point on the tangent sphere that 'touches' the hypersphere, if so, what is it?

2) If the Quaternion describes a specific point on the unit hypersphere, how can we logically interpret it's inverse. E.g. How would we do this with a 3 dimensional sphere, what would the inverse of a point there be? If I understand correctly the inverse 'P of P is defined so that P * 'P = I where I is identity, what in this case is Identity?

I hope my questions make sense, since I have no idea whether my assumptions are true. Anyway I would love some input on the matter. Fascinating topic this is. Thanks alot!

I have my own Unreal Engine branch where I implemented Dual Quaternion math for mesh skinning (move bones and the vertices follow). Dual Quaternions also help retain volume. With matrix interpolation, you get the candy wrapper effect when you twist (center between two bones goes to a single point). With Dual Quaternion, the rotation retains its volume (points rotate around a cylinder for tube like twist). Unfortunately, it tends to bulge for bends. I understand the math, but I still don't fully grasp why quaternions work when rotating a point.

The explorable video is so cool!

Omg so this is why... CFrame is a CFrame in Roblox.
Damn, amazing

uuuuuuuuuuuuuuuuuuh im gonna freaking expllode

stop

As far as I understand it correctly is a blackhole a 3d representation of a 4d object. i • i would switch the entire universe inside the border of this black hole, and all what was inside the black hole would be an entire universe outside. ☯️

Apart from the explanation, I adore the "Normandy Bridge map" music you use.

I will exam on Analytical Mechanics tomorrow and getting stuck to make sense of the orientation of rigid body with quaternions! Thank you so much for the video! Really very helpful!

Can someone pls tell me which software is this ?

Man, you guys should start a website that does that, heck i'd totally pay for access. Beats the crap of every alternative i've seen

How come in the interactive video in the 3D mode, in the angle mode, when the rotation is 45 degree, and the coefficients of i and j are 1 and 0 respectively, the three circles in the project are the same size only when the angle in the sine and cosine is 53 degrees?

why the conjugate of a complex number is shown as q^{-1}?

Thank you. Is there a way to get the quaternion that gives S times more angle of rotation than the original without using slerp (which is clamped to 0.0-1.0 in DirectX12)? I have a quaternion and need to have the rotation about the same axis, but S time more in angle.

Question:

How many degrees of freedom an object can manifest at the same time? For example, a sphere is spinning clockwise and moving forward at the same time, i.e. it’s using 2 out of 6 degrees of freedom, can it have more?

This interactive experience certainly helps a lot with such a complex topic. But one thing really missing I feel is some sort of conclusion as to why this is used for 3d rotations in computer graphics. The final piece of "rotate points in 4d space and then rotate them back as to not deform them leaving us with double the rotation" needs to be highlighted a bit more I feel.

Thanks Grant for this, I badly needed this

I know this is an old video, but I'm trying to learn about the standard model in terms of clifford Algebras, quaternions, and octonians. Any resources for those would be greatly appreciated. (I've already watched most popular videos, numphile, etc.)

Geometric algebra is better than quaternions

my brain is melting

After smashing my head against quaternions for almost a decade, this interactive simulation video has finally put things in some perspective.
Furthermore, the format of that platform is so powerful and SO smoothly executed.
Bravo, Grant and Ben. 👏 I am in awe!

It is not possible to calculate in 3D with a Pi based on 2D. Neither the Earth nor the Universe is flat!
Pi=3.14460551102969... 4/sqrt(ϕ)

my thesis will be using quaternions. Thanks for these videos and the website. Very helpful. I went from no understanding to a decent grasp of the concepts in no time at all.

Is there a link to the interactive demo?