The rotation problem and Hamilton's discovery of quaternions IV

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The rotation problem and Hamilton's discovery of quaternions IV | Famous Math Problems 13d

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Күн бұрын

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn [or half-slope--I have changed terminology since this video was made!] instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
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Пікірлер: 40

@MrMrodriguez12 10 жыл бұрын

That diagram on page 3 is incredible. Clearly this guy is a genius who has been trying to visualize these things for a long time.

@markptak5269 10 жыл бұрын

As always I am forever grateful for your incredible passion and vision. You're pretty radical for guy who doesn't believe in radicals.

@pierre-marcshinkaretzky8851 4 жыл бұрын

Magnifique ! I was amazed to see that it is possible to consider complex composition of rotations without angles. So nice course. So clear, so simple even the concepts are bright. Thanks professor! You are a wonderful teacher.

@aniliitb10 9 жыл бұрын

I just finished all your videos of this series. It took time and patience but yeah, I loved it. The last example was excellent, summed up everything about quaternions and added more clarification. I owe you sir!

@redemptionNFFC 8 жыл бұрын

Thank-you so much for these lectures - really built up from the ground, with great examples

@7steelrainbow 8 жыл бұрын

This is the best video explaining quaternions. From the first video of this series to this last one, the professor explains every little math-things needed to eventually understand the quaternions' rotations in 3d space. Even though I haven't done any math for a long time since high-school days, I could follow up this lecture series with some basic algebraic exercise. He literally builds up explanation from the ground. His way of teaching is brilliant.

@njwildberger 8 жыл бұрын

Thanks! Hope you watch some more.

@7steelrainbow 8 жыл бұрын

Sure I will, professor! Thanks to your great lecture, I could find how to deal with quaternion functions in my game project.:D I'm gonna watch through all your great videos whenever I can as I found that other lectures must be also extremely helpful for me. Wish I had found you sooner... Anyway, thank you professor!

@DcCornnnnn 7 жыл бұрын

You have expressed what I felt after watching these videos haha. I really like the way he teaches.

@7steelrainbow 7 жыл бұрын

DcCornnnnn Yes, I really like the way he teaches, too! :)

@acerovalderas 11 жыл бұрын

Another intellectually rewarding presentation, so clear!

@jehovajah 11 жыл бұрын

Excellent. The model of Quaternions is much clearer in this presentation. The redefinition of the trig ratios is also much more accessible.

@andreasmiller5448 5 жыл бұрын

Wow, you are a very good lecturer. Thank you for donating your time and effort to teaching us. I am interested in using quaternion mathematical structure to perform rotations in higher dimensions, as purely an academic exercise. There is one thing that I noted in one of your previous lectures that I thought was an error. There is an analogue to the cross product of vectors in more than 3 dimensions. The cross product of those higher dimensional vectors is equal to the determinant of a square matrix composed of vectors... one of the lines of the square matrix would be the basis vectors (I, j, k, w) in R4 for example, and the matrix would contain 3 other R4 vectors. The result would be a vector that was orthogonal to the first three. This makes sense intuitively, because in order to get an orthogonal in R4 you would need to supply 3 vectors. In R3, you would only need to supply 2 vectors.

@AugustoMathiasAdams 7 жыл бұрын

I'm loving using quaternions in robotics, instead of euler angles... thanks for the lectures!!

@busbug5457 Жыл бұрын

I appreciate you. This is an awesome video.

@mios6351 7 жыл бұрын

God bless you. I've failed several times to follow other materials explaining Quaternion which were introducing dry text of formulas and daunting matrices often without a 'teacher's voice. It was a pleasure to understand how the simple (but surprising) effect of multiplying a complex number represented on a 2D plane plays the key role in the 4D Quaternion's multiplication acting as rotation. Your explanations were easy to understand. Thank you again for the series.

@njwildberger 7 жыл бұрын

You are most welcome.

@njwildberger 11 жыл бұрын

Thanks so much for catching this! I must have been dreaming in my calculation at the bottom of page 8. I will fix this and repost sometime shortly.

@brendawilliams8062 3 жыл бұрын

Thankyou.

@heruilin 11 жыл бұрын

Well done!

@jaanuskiipli4647 6 жыл бұрын

Good stuff, thank you!

@njwildberger 10 жыл бұрын

Quite correct.

@MetalManiacBoy 3 жыл бұрын

THANK YOU SO MUCH

@auspicious99 11 жыл бұрын

awesome, thanks!

@kolboch 5 жыл бұрын

Glad I got it, else would think that every time you smile you are smirking at me :D

@stefandonchev7918 10 жыл бұрын

Double Like! Thanks a lot.

@pickeyberry4060 2 жыл бұрын

Video Content 00:00 Introduction 05:54 Half-turn of a vector 17:19 Multiplication by a unit vector 27:05 Multiplication by a General quaternion 39:17 Quaternion rotation theorem

@MrParthab 5 жыл бұрын

Sir, these lectures are very helpful for me. I am working in space industry, where spacecraft attitude we are representing by quaternion. While using in control system we are converting quaternion into angle, where we are using q=(t,v) = (cos(d/2), sin(d/2)). Where d is the composite rotation. Can you give some geometrical interpretation of this formulation?

@peepzorz 5 жыл бұрын

Thank you for the wonderful presentations. It seems that half of a full rotation (aka 180 degrees) is a problematic singularity with quaternions, causing the half-turn to be infinite? Do you have suggestions for how this case is dealt with?

@njwildberger 11 жыл бұрын

Excellent. Just remember I made an arithmetical mistake on the second last board, so the results need to be reposted, which I hope to do sometime in the near future!

@alfonsovanacore1139 Жыл бұрын

The mistake is h=y/(x+r) on board 9:31? Thank-you so much for these lectures

@Suleimanmohammad1977 8 жыл бұрын

your lectures are really great. did any body tell you before that you look very much like Steve Martin?

@pythagorasaurusrex9853 7 жыл бұрын

Thanks for that great lecture. I never clearly understood, why one uses the multiplication z*q*z_bar. I saw this famous formula several times before, but nobody clearly derived this nor explained it to me.

@njwildberger 7 жыл бұрын

You're welcome, glad the idea is clearer now.

@mechwurm 11 жыл бұрын

thanks for this. Better than buying a book (which Ill probably do anyway)

@iangrant9675 5 жыл бұрын

So a half-turn is a unit of one right angle, which is the Euclidean measure of angle: all right-angles are equal, ...

@njwildberger 10 жыл бұрын

These principles are fine. I just don't want to engage in endless discussions/ arguments which come down to the meanings of words. I am trying to find a way to set-up mathematics so that most of the philosophizing is removed.

@jaanuskiipli4647 6 жыл бұрын

If Q(z) = con(z)*z then why not simplify the formula phi_z(q) = z*q*con(z)/Q(z) to phi_z(q) = z*q/z ?

@njwildberger 6 жыл бұрын

Yes you can do that, but it is not really a simplification. Usually with a quotient of complex numbers we multiply top and bottom by the conjugate to simplify.