Quaternions and 3d rotation, explained interactively

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<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption authors (Spanish)</h4>

<div id="content" class="style-scope ytd-metadata-row-renderer"><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UC8tic89FnDFfRCNRHIEQY6w">MATECLIPS</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCIgANqQeTg3llfnibt0gj8w">Javier Wagner</a></div>

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<h4 id="title" class="style-scope ytd-metadata-row-renderer">Category</h4>

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SHOW LESS</div>

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<div class="cue style-scope ytd-transcript-body-renderer active">In a moment, I’ll point you to a separate website hosting a short sequence of what we’re</div>

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<div class="cue style-scope ytd-transcript-body-renderer">calling “explorable videos”.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It was done in collaboration with Ben Eater, who runs an excellent channel about Computer</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Engineering which viewers of this channel would definitely enjoy, and all the web development</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that made these explorable videos possible is thanks to him.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I don’t want to say too much about it now, it’s really something that you have to experience</div>

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<div class="cue style-scope ytd-transcript-body-renderer">for yourself.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Certainly one of the coolest things I’ve had the pleasure of working on.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So to set the stage, last video I described quaternions, a certain 4-dimensional number</div>

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<div class="cue style-scope ytd-transcript-body-renderer">system that the 19th-century versions of Wolverine and the old man from Home Alone called “evil”</div>

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<div class="cue style-scope ytd-transcript-body-renderer">for how convoluted it seemed at the time.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And perhaps you too are wondering why on earth anyone would bother with such an alien-seeming</div>

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<div class="cue style-scope ytd-transcript-body-renderer">number system.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">One of the big reasons, especially for programmers, is that they give a really nice way for describing</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a 3d orientation which is not susceptible to the bugs and edge-cases of other methods.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I mean, they’re interesting mathematically for a lot of reasons, but this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is probably their biggest practical application.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">To take one example, a friend of mine who used to work at Apple, Andy Matuschak, delighted</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in telling me about shipping code to hundreds of millions of devices that uses quaternions</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to track the phone’s model for how it’s oriented in space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">That’s right, your phone almost certainly has software running inside it that rely on</div>

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<div class="cue style-scope ytd-transcript-body-renderer">quaternions.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The thing is, there are other ways to think about and compute rotations, many of which</div>

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<div class="cue style-scope ytd-transcript-body-renderer">are way simpler to think about than quaternions.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For example, any of you familiar with linear algebra will know that 3x3 matrices can really</div>

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<div class="cue style-scope ytd-transcript-body-renderer">nicely describe 3d orientation, and a common way many programmers like to think about constructing</div>

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<div class="cue style-scope ytd-transcript-body-renderer">one of these matrices for a desired orientation is to imagine rotating an object around three</div>

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<div class="cue style-scope ytd-transcript-body-renderer">easy-to-think-about axes, where the relevant angles for these rotations are commonly called</div>

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<div class="cue style-scope ytd-transcript-body-renderer">“Euler angles”.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This mostly works, but one big problem is that it’s vulnerable to is something called</div>

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<div class="cue style-scope ytd-transcript-body-renderer">“gimbal lock”, where two of your axes of rotation get lined up and you lose a degree</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of freedom.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It can also cause difficulties when trying to interpolate between two orientations.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">There are many great resources online for learning about Euler angles and Gimbal lock,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and I’ve left links in the description to a few.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Not only do quaternions avoid issues like gimbal lock, they give a very seamless way</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to interpolate between two three-dimensional orientations, one which lacks ambiguities</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of Euler angles, and which avoids the issues of numerical precision and normalization that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">arise in trying to interpolate between two rotation matrices.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">To warm up to the idea of how multiplication in some higher dimensional number system might</div>

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<div class="cue style-scope ytd-transcript-body-renderer">be used to compute rotations, take a moment to remember how complex numbers give a slick</div>

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<div class="cue style-scope ytd-transcript-body-renderer">method for computing 2d rotations.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Specifically, let’s say you have some point in 2d space, like (4, 1), and you want to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">know the new coordinates you’d get after rotating it 30 degrees.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Complex numbers give sort of a snazzy way to do this: Take the complex number 30 degrees</div>

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<div class="cue style-scope ytd-transcript-body-renderer">off the horizontal, a distance 1 from the origin, cos(30o) + sin(30o)i.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now just multiply this by your point, represented as a complex number.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The only rule you need to know to carry out this computation is that i² = -1, and in</div>

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<div class="cue style-scope ytd-transcript-body-renderer">what might feel like a bit of magic to those first learning it, carrying out this product</div>

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<div class="cue style-scope ytd-transcript-body-renderer">from that one simple rule gives the coordinates of a new point, rotated 30 degrees away from</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the original.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Using quaternions to describe 3d rotations is similar, though the look and feel is slightly</div>

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<div class="cue style-scope ytd-transcript-body-renderer">different.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Let’s say you want to rotate some angle around some axis.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">First, define that axis with a unit vector, which we’ll write as having i, j and k components,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">normalized so that the sum of their squares is 1.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Similar to the case of complex numbers, you use the angle to construct a quaternion by</div>

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<div class="cue style-scope ytd-transcript-body-renderer">taking cos(that angle) as the real part, plus sin(that angle) times an imaginary part, except</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this time the imaginary part is the 3d axis of rotation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Well, actually you take half that angle, which might feel totally arbitrary, but hopefully</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that will make some sense by the end of this whole experience.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now let’s say you have some 3d point, which we’ll write with ijk, components, and you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">want to know the coordinates of what you’ll get by rotating this point by your specified</div>

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<div class="cue style-scope ytd-transcript-body-renderer">angle around your specified axis.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">What you do is not just a single quaternion product, but a quaternion sandwich, where</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you multiply by q from the left and the inverse of q from the right.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If you know how i, j and k multiply amongst themselves, you can carry out these two products</div>

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<div class="cue style-scope ytd-transcript-body-renderer">by expanding everything out, or more likely by having a computer do so.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And, in what might feel like a bit of black magic, this big computation will return for</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you the rotated version of the point.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Our goal is to break down and visualize what’s happening with these two products.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I’ll review the method for thinking about quaternion multiplication described last video,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">explain why half the angle is used, and why you multiply from the right by the inverse.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">On the screen, and at the top of the description, is a link to the website Ben Eater setup with</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the explorable videos.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It’s...it’s just really cool, Eater did something awesome here.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">At the very least, you should go take a quick look, but I’d love it if you went through</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the full experience.</div>

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Quaternions and 3d rotation, explained interactively

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