The rotation problem and Hamilton's discovery of quaternions (II)

The rotation problem and Hamilton's discovery of quaternions (II) | Famous Math Problems 13b

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This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Video Contents:
00:00 Introduction to rotations and their composition-
03:25 Rotations of 3-Dimensional space ( geometrically )
09:38 Planar situation
14:25 Algebra of planar rotations
23:36 Rotation of 3D space as a product of 2 reflections
37:04 Algebra of 3D Rotations about 0
42:57 Euler theorem - The product of two rotations is a rotation
44:40 The analytic approach ( via Linear Algebra)
48:24 How to define 2 directions to be perpendicular; vectors
52:43 Cross product of 2 vectors
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Пікірлер: 29

@RugnirSvenstarr 2 жыл бұрын

As a robotics engineering student im bloomin glad Hamilton figured this out because we'd never get anywhere without it 😁 I've seen servo motors worth over 1k dollars damaged because somebody forgot to use quarternions instead of euler angles, resulting in extremely rapid angle changes

@aniliitb10 9 жыл бұрын

i have never had such excellent lectures for free!

@robharwood3538 2 жыл бұрын

Hey Prof. Wildberger! Re-watching this (and many of your other videos) gave me an inspiration for an alternative name for the so-called 'Real' numbers: *Astral* numbers! 😃 Here's the reasoning for the name: In this video you talked about the 'transcendental' nature of angles, and that angles are originally motivated by Astronomy. Since 'Real' numbers are not something we can actually 'obtain', it's as if they are 'very very far away' (as in, you can 'approach a limit' but never actually get there). In this way, 'Real' numbers are more like very distant objects in our universe, such as very distant galaxies, or even 'objects' that are actually *beyond* our observable universe (completely inaccessible to us!). The term 'astral' originates from the same origin as Astronomy and also Astrology, namely the 'stars', very distant 'points' in the sky. These days, it more so has a connotation of as belonging within the imagination, or as a form of 'superstitious' (or, more charitably, 'spiritual') belief in a 'reality' that is separate from the actual physical world. E.g. many 'spiritual' folks these days talk about 'astral projection', which I interpret as either being perhaps a deep form of day-dreaming (if done while waking), or more probably as a form of lucid dreaming (if done while asleep). While we can 'imagine' and 'talk about' our 'astral experiences' with so-called 'Real' numbers, they do not actually correspond with actual-real world calculations: we cannot actually demonstrate them concretely, _outside_ of our imaginations. So, we no longer want to elevate them as actually-real with the confusing term 'Real' numbers. Instead, we'll call them Astral numbers, to reflect their origins (attempts, like the early theories involving never-ending epicycles-on-epicycles, to understand Astronomy), while also reflecting the more 'imaginative' and 'beyond the universe' connotations of the word 'astral'. So, when talking about complex numbers, we can perhaps still talk about the 'real parts' and the 'imaginary parts', but when talking about *transcendental* notions we'll refer to those as Astral numbers and Complex Astral numbers to distinguish them from Rational numbers and Complex Rational numbers (with Rational real-parts and Rational imaginary-parts). [Or, perhaps, another idea I had regarding complex numbers/naming: The 'real' part should be called the scaling or scalar part and the 'imaginary' part the 'transmorphing' or 'morphing' part (rotating for Blue, reflecting for Red and Green). Or, maybe just the 'vector' part, if that's more palatable. Scalar and vector are what's used in Geometric Algebra, so maybe that's easier to adopt.]

@dr.rahulgupta7573 4 жыл бұрын

Excellent presentation of the topics. Thanks a lot. DrRahul Rohtak India

@gustavnordin8690 3 жыл бұрын

wow - i have some ground to cover yet you made it very understandable

@chasr1843 6 жыл бұрын

This is one of your Geatest lectures Dr. NJW

@njwildberger 11 жыл бұрын

Hi, it turns out that there are relativistic variants of complex numbers, and also of quaternions, that help us in understanding rotations in a relativistic plane and in the hyperbolic plane. I might discuss these issues in my WildLinAlg series at some point, of my UnivHypGeom series. It is a good question!

@harshverma6425 8 жыл бұрын

Thank you so much , I had been struggling to find a free online source that could teach me rotations in "detail". I have always been a person who craves a complete understanding of both the algebra and visual components of math concepts. Moreover, your series is a bonus series because I am also learning about rational mathematics.

@dsfgoppudfgihdsf 5 жыл бұрын

So good.

@notcrazyyet 4 жыл бұрын

wow! thank you!

@patrykbochenek 10 жыл бұрын

I really like how you explain the dot product @51:18 When I was taught this at school, it was done exactly in the manner that you described, that is: "Here is the definition of dot product. Period."

@Jack-sy6di 10 жыл бұрын

I remember being incredibly confused that we called it a "product". What seemed fundamental to me was the identity xx' + yy' = |x||y||cos(theta), and it seemed more natural to simply state that theorem rather than define this bizarre operation. My favorite argument at the time was: When we studied the Pythagorean theorem, we didn't define the "Pythagorean product" of two numbers as being the square root of the sum of their squares. So why does this formula deserve to be thought of as an operation, rather than just a formula?

@njwildberger 9 жыл бұрын

Jack Eiler If you check out my WildLinAlg series, the second half, you will see that the dot product and its various generalizations are the basic way of introducing metrical structure into linear algebra. So the dot product allows us to DEFINE notions of perpendicularity, rotation, similarity etc. It is a very important idea!

@VivaNadia 3 жыл бұрын

Thank you so much for the awesome lecture series, it helped me to see the problem that I am dealing with different perspective

@7steelrainbow 8 жыл бұрын

Great! Thanks for this wonderful lecture!

@paul1964uk 11 жыл бұрын

Another great lecture. Thank you. This approach of building up concepts gradually before introducing/re-evaluating the main historical object (Quaternions in this case) is useful on so many levels as it sheds a lot of light and sets a direction we can follow, and greatly enjoy doing so. btw the year (of discovery) in the description should read as 1843 (that's been bothering me for some reason!)

@brendawilliams8062 3 жыл бұрын

Thankyou

@lidorshimoni5470 2 жыл бұрын

YOU ARE GREAT. alot of time i thouth what happend "meanwhile" we moves on point to another...

@MarcoAurelioCurado 9 жыл бұрын

thank you for the lesson!

@marzioforte9364 11 жыл бұрын

Hello, luv the vidoes. Looking forward to 13c.

@njwildberger 11 жыл бұрын

Yes, still to come.

@phpnepal 10 жыл бұрын

Thank you sir! That helped a lot!

@jauhueitang6879 2 жыл бұрын

Excellent presentation!

@STIVESification 8 жыл бұрын

Thank you !

@teoaero 11 жыл бұрын

great vids, helped my 3d understanding lots! I add my voice to the clamour for part c! :)

@pickeyberry4060 2 жыл бұрын

Video Content 00:00 Introduction 03:24 Rotations of 3D space(geometrically) 09:38 If rotation axes of subsequent rotations are close together, this is approximately a planar situation 14:25 Algebra of planar rotations 23:36 A rotation of 3D space is also the product of two reflections: about 2 planes through origin 37:04 Algebra of 3D rotations about 0 42:57 We have shown: Theorem (Euler 1776): The product of two rotations is a rotation 52:42 Cross-product of 2 vectors