Bob Palais - Quaternion multiplication as geodesic vector addition on S2 - JMM2018 Quaternion

Transcript

and then come back to the and then start the what I the PowerPoint thanks for coming that I figured I'd since I was changing programs and I have a screenshot this so you can see going from 2 to 0 yep back to to the other way and forward so there's a fell trick and you can instead if you prefer I think this will undo play trick instead there is quite trick with a fixed shoulder generated from the same color toffee this will be over and it's about 2:30 in and I'll start but they just be nuisance that was in there I would start with that we can all keep your 3d glasses and download this free program that has all kinds of cool stuff in it like even things like space curves and tourist knots and anaglyph 3d there's your tourist notes and you can change the number of wraps and you can view them as a tourist and they'll always tell you stuff like set parameters 101 and instead of 5 and to me was to the 101 23 Torrez gone [Music] like oh wow anyway on to the higher this is a great program that was originated by my father who's in the audience so but and there'll be one more 3d so keep your glasses ready and we will start the show hopefully for you from current small and I think I want to start with the announcement is solved the hood Ian literally it's still happening if you put in Riemann hypothesis in Google as you will see me do right here you will see Riemann hypothesis those zeros except for real part 12 well well I just decided to go over a Wolfram Alpha and put in it once with real part 1/2 that is oh my gosh for example Oh anyway that's it so we'll start out with a counter example to the Riemann hypothesis literally the Riemann hypothesis according to Google and it really happens try it it's amazing um and what else about that thanks to Andrew the Wisco for calculating those euros actually his first one um so I actually got into this return Ian thing when I was also working on tumour suppressing therapy just like meet her to talk about yesterday and last year which was so exciting and the differential equations of cell proliferation and suppress cell death and we looked at the data this was in 2002 early precision medicine looking at the process meter of the cells that's really what a gene expression our array is it says how many times our protein which is a program from a gene which is a program for a protein is called and we identified three genes before the between the dormant and aggressive transition just in the numbers and I said well they're never going to be able to test whether I know what I'm doing and they said those three genes are on a pathway and they say just like you said that cell death is suppressed and and proliferation is accelerated they put into the words the gas pedal is stuck and the brakes are broken and they said we can affect that and by by by affecting that they actually suppress tumor growth when fungal growth was inhibited using something we came out of man how did I get it I had it they said can you explain how you got the numbers while I was looking in various individual genes pairs of genes triples of genes and there was one triple gene in the center of 7,000 that separated the good from the bad in 3d but in any two dimensions any two gene projection they overlapped like this and kocho the weed scientists pathologist said that would be really sexy who couldn't move those in 3d and I knew about virtual track calls but I've never done one myself so I had to go to the family member with programmed virtual track balls and help me do this and so there it is you can actually see their growth anti a popped up photonic effects and growth-promoting the breaks are broken the gas pedal is stuck before and after and this was really there in activation of this pathway inhibits tumor growth it was amazing to me math and geometry could show this and so I programmed my first virtual trackball that is actually referred to in the program the little black balls was all I could figure out to make axes so I have a plus e 1 and a minus e 1 and if you all fine things along one axis its overlapping and it will align on any of the three axes its overlapping there's another axis I'm gonna line up early flash ActionScript there they're lined up still they're overlapping but in 3d if you do it right from a certain perspective in 3ds it really shows that they're separated by those genes and that's why I showed them they said painless let's play this and see if we can get a therapy and it worked so quick review of or introduction if nothing else this might be fun a very good thing for in teaching classes could exercise in vector geometry how do you actually render those points in 3d onto a screen ok I'll tell you that you only really need two unit vectors to determine everything because you really want you one unit vector here's my data point in r3 here's my camera that's all I need to determine the point on the plane through the origin orthogonal to my line from the camera to the origin where the data point intersects as perspective drawing all I need is one vector this this camera position that tells me where the interpolation I'm going to talk about interpolation later interpolation from the view point to the data point that's a line that's this formula VP plus T times DP his VP data point viewpoint um all you need to find is the T that makes that orthogonal to V P well I take the dot product solve the equation there's my teeth done one vector gets me the point in the image point but that doesn't get me the point on the screen I need one more I need to know what up is why do I just need up because if I know my direction towards the screen and up yeah I can do a cross product than you appeal and get my record and so there up dope and all I really need is since I know the distance if I just keep the distance from the camera to the origin the same all you need is a unit vector that way so I'm gonna have a orthonormal frame that's defined by two vectors so I need my up and my forwards and I can cross them to get my right and I can scale my fixed constants to get my pixels and my camera distance so how do I find that's it already same thing the image point to get points my coordinates I just need my basis in the in that orthogonal plane and take the dot product with this and this and then convert to pixels how do I need know that I need my up vector and I get my other vector from that so to literally to draw a point in space on the screen correctly all I need is two vectors it completes it to an orthonormal frame okay then how do I go that's the summary of that aside from the viewing distance and scaling factors all you need to render points Part B forward and deal the third is obtained it by the right give the cross-product okay and this is what you get out of that when we played it [Music] like play there okay so there's the ten-second so I did a tetrahedron the vertex will follow the mouse and it's passed the Technic notice when I pull to the right and back up purple ends up on the right when I started to get started again and pull to the left and back up purple will be on the other side so it is path dependent but it's following my my my mouse so see the the past dependence and following the mouse later I'll show you an animation of Schumacher's arc ball the quaternion ik virtual trackball I implemented that the same and I will show you that it is not path dependent and then it doesn't follow the mouse it doubles the angle so one Paul can flip it over that's either a feature or a bug depending on whether you now we kill them anyway so this is what you get so how am i rotating things with the mouse do I have am i rotating the points of this tetrahedron based on where the mouse is maybe that's what I want people to do a mouse point specifies a unit vector again in the interpolation formula here is a point that I convert from its pixels to its point in this plane and then I just do that interpolation again but instead of looking for an orthogonal I look for norm squared equals one a couple dot product you got a quadratic equation sometimes it will intersect in two points one in front one in back sometimes it actually intersects in two points in front you gotta watch out for the horizon that's actually the trickiest part about implementing it otherwise things your mouse will fall off the horizon that's what I discovered so when you're interpolation the crystal ball unit sphere that's all human is finding unit vectors from my mouse so a mouse point specifies that oops um so two mouse points and I think I'm there it says to my sports the plural Mouse motion gives you two unit vectors that's gonna specify a rotation right the to two unit vectors specify a three dimensional rotation uniquely uh-huh you could pre rotate about this axis post rotate about that axis still takes this to this not specified uniquely until you say it weaves the orthogonal complement alone in this three dimensions it preserves the orthogonal the the cross comment there so if you one is you - or - you - what rough um it's it's the identity is the identity rotation takes it to itself but anything wrapped around it so you need more specification preserve the orthogonal complement preserve the PI identity so now here is the fun part it's all fun how do you apply that rotation from your to unit vectors specified by the two unit vectors it takes the first to the second and preserves the axis the orthogonal how do you apply that rotation to make the image rotate a lot of people actually rotate all the points of the tetrahedron or the icosahedron or the 120 thing oh but it turns out much simpler to reverse rotate directions of your viewing frame don't rotate them so anytime you see something rotate on the screen today you're never that thing is rotating at all it's just you're moving the camera backwards so that makes it pretty fast okay how do you apply our deform what's the factor like like what's the what's the factor did you scale to 203 some numbers here yeah so how do I actually apply that so the idea is still um first of all oh oh actually no your question the factor of two vectors versus a lot I saw a actually a workshop in one of our our section meetings that said isn't this amazing that we can uh I guess nowadays you can brute force man perfect but their words it we're rotating thousands of points with the mouse it's like I guess yeah but you don't have to it's a single object this is a single object world and if you have multiple objects I've realized if you have multiple cameras to render each one independently you can still see the same thing yeah it isn't stuck to a single object world you just have multiple cameras thank you dick that you will see that when I do this string trick the tangle trick I had to use two cameras because I wanted the sphere to rotate and then the strings to rotating and that was two cameras on the same screen um so um so first thing yes much better use to the reverse rotate the camera second thing to generate that actual rotation to do that well a lot of people most people actually still do the graphics textbooks all say cross-product find the axes turn around and treat some transcendentals even worse than those square roots right you're absolutely right and we're gonna be talking about rationals and I like that's why I think I / oh maybe up ahead I divided by the norm square if not the square root I don't like square roots any better than you do so arc signs or our cosines or blowhard Rodriguez matrix and you'll find the matrix from the axis in the angle or the geometric axis angle formula they're actually the same or you could do the quaternion way the quaternion way you can find the cosine of theta over two that might actually be involved a little bit even if it isn't even if there are other ways to get cosine theta over to inside in order to to do a Hamilton's conjugation of a quaternion to get a rotation the first one is a pure vector only three nonzero components 3 times 4 12 but then you got four components and then four times three four times for another 16 you got every a quaternion conjugation is 28 operations matrix times vector 9 matrix that's why Boyer's formula coming up to go from s 3 there is no 3 to get quaternion matrix at what were had you didn't have the composition but he definitely had done quaternion 2 matrix algorithm as much thing so you could contrive to construct that use what I call weight where shoot monkey s3s m3 quaternion 2 matrix plus million vectors to multiply and people to do since they rotate the points themselves so there are lots of bad ways to do it and in fact there's I'll show you a picture from a google Tech Talk from Invensys all about sewing how great quaternions are and they say find the angle and then put it into boiler rotate make the matrix they miss the quaternion two matrix things so a lot of people miss understand that is a good thing okay here well he's more into linear algebra I guess orthogonal reflection across a subspace generally you can decompose any vector in our n into a vector in a subspace and orthogonal to that subspace uniquely right okay reflection across the subspace just a number that orthogonal partner would be plus e to the minus e here I'm saying C times V because I'm going to take a unit vector sometime and so that's what how do you get from v+ v- e it's you double twice V Plus E and then minus a B and you get B minus e I think over to here it is so here's the beauty here is reflection across a unit vector is a reflection is orthogonal Pythagoras perpendicular it preserves lengths and angles a reflection is an orthogonal distance preserving transformation but why is this in a rotation always because it might not be orientation preserving it's minus 1 on the orthogonal complement so it depends on how many dimensions there are if you reflect across a line in three dimensions you have two orthogonal dimensions that's 2 minus 1 to the 2 is plus 1 and I reflection across a line is actually a rotation 180-degree rotation around that line you'll see this in a second because all pictures there are orthogonal projection I want to break something down into a multiple of B plus a residual E and I want that residual to be orthogonal that derives immediately the formula for orthogonal projection onto a vector many gram-schmidt can do it term by term but that's how I like to let students divide just say I want something purpose along and perpendicular never has to be perpendicular to this we just it pops out so here now we get back to pictures what does this have to do with return well how do you in reflecting across a line is a rotation itself but reflecting across Toula aligned twice no matter what the dimension any reflection twice is a rotation because it's if the first one was an orientation preserving twice it will be okay so here's the cool picture cool animation now if I want to rotate the plane containing these two vectors from u 0 from noon to 4 o'clock how do I do it I also just use that dot product and reflect across u 0 reflect across moon and reflect across 2 o'clock reflect across your first unit vector and reflect across the Sun or the bisector and let's see if that works so first I'm going to see that the reflection I'm going to play this movie you're going to see that this reflection is a rotation so I'm gonna rotate around and see that there now we rotate around two o'clock what did I get I rotated that the watch from 12 to 4 twice as far that's two reflections is rotation now I'll do it fast I won't rotate around them to season prophecy I really cross that not only reflect across that that rotation by twice the angle there is reflection by across two things is is a reflection is a rotation of it now I always said no doesn't the same argument say that reflection of cross two o'clock takes this to this and reflection across the final vector fixes the orientation so yeah it does so you can do this first and by sector or by sector then second and hey if it's those two white not three o'clock and five o'clock reflect across three o'clock reflect across five o'clock what do I get I still got the four hour rotation to me that looks like an equivalence class all ordered pairs not only that I could reflect across this or I could reflect across minus that and get the same rotation that's a unit quaternion an equivalence class of ordered pairs of unit vectors in r3 sharing the same dot product and cross product and the rotation you don't do Hamilton's conjugation you do two reflections across the two vectors and you get the double ankle rotation and that's a great algorithm but it shows so much more first of all I immediately see that if I reverse one of the vectors I get the same reflection right reflection across this is the same as reflection across this opposite what happens in the dot cross product if I negate one of the vectors they both get painted the scalar and vector part are negated but I get the same rotation that's the negative quaternion that rotation aha so I want to understand all other things from this point of view replace trackball math this in Vincents says do the arc arc tangent and plug it in so people don't do this this is Channing's original trackball patent doesn't do this oh here's shoe monkeys aren't artful and fun to see this is a contrast this is good turn Yannick so you see the double angle and it's path independent see the arc it even shows the arc on s2 and nice thing nice thing in a way is you can do the full flip with a half rotation but it's totally passive independent because it just keeps the original of what you put down watch I can flip it over and full turn with a half turn yeah so that's shoe monkeys are bone so it's different an equivalence class of ordered pairs I learned about in the and equations class can we compose rotations by the same construction we use to compose translations if reflect across this row vector cross this is one quaternion reflect across this reflect across this is another how when I compose them into two into a representative another equivalence class any two planes in r3 meet so I can find that what happens to those two reflections across the common unit vector they cancel their you get the composition in which case quaternion multiplication we suspect is the relationship between dot and cross product of any three unit vectors in r3 if I know the dr. cross-product Avista dot cross product of these two the dr. cross product of these two should be related as quaternion multiplication that came out with breakfast and we interpolate rotations the same way how do you interpolate two translations instead of having the tale of the first be the head of the second you have them both have the same tale and interpolate the heads so I'm just find the same common starting vector and interpolate I've reduced s3 interpolation to s2 I can do return an interpolation by doing an arc on literally s2 in our three instead of s3 in our four it's faster can we go in the other way gee this double reflection thing is really nice does that work back in translations what if I reflect across the tail of the vector then reflect space points in space everything goes to its mirror image across the point the head of a vector we did translation by twice going through the vector the analogy works perfectly if I reflect across this point in space and reflect across this point in space I end up this point goes to support no it all goes through and finally can we understand unity through this view can we understand unit quaternion to the two-to-one bamboo that's 3d asari mature you have an entangled trachoma toffee yes so here I actually made a concrete viewing a quaternion 'us pair of unit vectors here is this quaternion here is its perspective that here the two vectors and I can change the representative I am a button to change the representative I have changed them the gate and invert what's the inverse real white across this reflective crust reflect upon them across the first vector reflect across the second vector how would you invert that reflection cross this the first thing to reflect across the second how would you invert that first what happens to the dot product the same what happens the cross product if I in Reverse the two vectors and the game system the conjugation right negate the vector part there you know so I see quaternions explicitly concretely with respect to a basis see I can rotate my because I'm using this this actually generates all the one decibel unit quaternions one decimal digit exact one decimal digits by finding all the four integers whose sum of squares is this water and then divide by up to 10 it's kind of fun numerical number theory okay so negate inverter I can show you all this - reflections doubles translations and wind rotations so got no home it'll be denied where is it it's not there it is yeah that's the vibe this came to be one telephone book this came to be one day wrapped in a thick oak says if you read this you'll understand why it's so heavy and in it is an amazing picture that says if you take a ball in the center of the room and knew this and so I also tried to make simulators and stuff with tape and pins that half the time it works half the time it didn't I don't know so I had to try and understand it better but here's the point okay be good if I reflect across a unit vector and reflect across the same unit vector what transformation of space do I have reflect across the same thing twice yes so if I take a path of a second vector on s2 that starts it begins at the same vector as the first reflection vector the start in the end will both be the identities if I do my second reflection on an equator that goes back on PI over two a quarter-turn PI over two they should really have a different thing that would make that a quarter what would you call it tau anyway you do a quarter turn down below the angle we already saw my double-doubles the angle if I do a quarter turn the rotation is how far if the reflecting vectors are a quarter turn apart how far apart are the that is the rotation after 90 degrees hundred eighty thirty sixty we saw the SS the o'clock two o'clock four o'clock so if I do my take my reflector one turn around the equator how far did the rotations go twice so now we just do the rubberband Homa Tommy young - whatever it is it's a it's a morph from two turns about the vertical axis to constant that the identity and he turns out to be David's beautiful home toffee he figured out that they are the same and so here what you see are in the past equator so another way to see so they're my past - - that generates about trip but that's the path of the second reflecting vector where the first remaining maker is always that and if they start to finish there it's very explicit again a good vector geometry they find the intersection of a plane with a spear a pointing at centered anyone see I don't think I'm taking the plane going like and I'm getting those paths and parametrizing them so here you get belt trick two full turns of the belt down here along the path of LV the edges are both the identity represented by a square image of a square and I'll rub it here's my belt going from two turns we'll go through the Homa toffee slicing this way and at the end you will get the flat belt from literal hope it's empty square at the plate drink his vertical slices and same thing there's the two turns my camera has clicked up so you can tap this beat two full turns one turn the two terms straight arm straight arm twisted arm after one turn and let it run there's your home and toffee version of me yours tangle drink and this is the one that mr. Foreman and viewer yeah you want us you can be the ceilings all this you translation but no rotation hold that give me slack if I heat it up a little bit let's get it flat there perfect okay now I'm going to do one full turn oops like to step on this okay ready one full turn two full turns give me a little slack that'll trick over that'll trick under fortunately and go around they chase each other around there it is on the top you'll and it's reflected it's twist this way to his family if you put twists in the head this way and in the same orientation it would work as one of the be trying to go this way what would be trying to go that way so that's for topologist that's one you will be able to explain why in why if the two terms go this way and the two terms go that way you can't do it without a collision there's an obstruction I I don't know so it's reacted so here's another way of doing it instead of the cubes why not rotate a reference rotation of a sphere not only a sphere concentric spheres along the path from an inner sphere to an outer sphere well then there's no way paths can intersect cuz it's rigid rotation at every radius if it's the identity it's just radial lines if it's two turns about a vertical they're gonna wrap around the sphere twice and the home atop is somehow gonna unwrap it okay I think we got to put the 3d you could put the 3d glasses when I can send you links to but there it is [Music] you have an outer sphere and the Internet's just literally the path of the home atop beyond the single point and on the top II and this is two cameras if things aren't moving the same and you can do you can do it two terms or four terms or eight terms so the eight Turk one looks pretty cool and you can change your review money you can actually do this in a virtual trackball and watch it from different angles so I can send anyone a link for that okay here doing them with a fixed point looking at that here are my two turns and the identity going across and this was passed in our three that represent the position our three represents the axis and the distance from the origin is the angle going this entry from 0 to 2 pi of infinity so the r3 plus the ball for s3 so that's that's an early Java representation good Bob isn't that happening in art actually is using in our two slices of our 3d axis I'm using an art to slice of our 3/4 they actually that's exactly that's your liver receiving the point and what you're doing is you're going from two turns which is that infinity to the origin so you're doing the reverse contour integral in making our go to 0 infinity as David noticed it's got it pauses for a long time while it's waiting to come back in the it's exactly what he was talking about yesterday about that going out this axis is the same orientation as going in the other X going in from minus infinity the opposite direction very beautiful that he came to and the and we could wave it yesterday and all the way through we have lots of correspondence before this and I've learned a lot more about my own time bifur so the origin of this was then what is the formula could we derive from the geometry the relationship if you know the dominant correspondent of this duck cross-product abyss can you get the duck respond with this so I was thinking what's the structure just like what is the structure of complex multiplication in the circle rotation rotation in the circle so notice that how I'll teach in class complex mobile even when the 0 goes to X Y then just turn your head this becomes 1 0 this becomes minus y X and then X 0 goes to X times x y y 0 goes to Y times minus y X add them up there's complex multiplication and some of the co-starring the dishes were moving the second addition formula done so I wanted something that would show the structure complex multiplication is x times X y plus y times minus YX capital and small this was an article in the notices in a little it so from that once you have that that that's rotations not only that this also gets you x squared plus y squared equals 1 because if one 0 goes to X Y X negative 1 it goes to 1 0 there's Pythagoras as theta minus theta equals one cosines in digital format so that's all stream on a tree in the can of the way so you can give the geometric interpretation by explicitly rotating to a standard configuration I can rotate any one vector so that that's all but the first coordinate is 0 second vector so all into the XY plane all with the second vectors the third vector I can make all but the first three coordinate of 0 and so forth that's QR really that's a Q are orthogonal triangular decomposition we get a proof of winners theorem from this actually Oh but so here I need a composer oh this is IJ equals K the equivalence class j j comes first I circle J so J comes first there's that's J dot product 0 cross product here oh one zero I will be dot product zero cross product one zero zero was the scalar part victor part and you compose them by that first of the second second on the first drop them out first student of the first second of the second there's K know that I made a much more interpolation interpolation works with the first vector is the same and interpolate on to show you here's interpolating quaternion geometrically common first vector second vector signature here the dot product is up is the crossbow does this and now I will interpolate that how do you interpolate you just interpolate the second vectors on s2 it reduces the dimension for interpolation when you view with turn the ends this way there are points so those will be interpolation on has three literally if you look at the four components of the squares it will be here is a composition concretely kind of a fun thing about this is so you're a few it's a little calculator thing that I've got there that you can take this off you can choose a scalar which are the end of elementary ijk plus or minus a vector with scalar from zero or a general one and you just click through and they'll it'll get them all so you can actually see quaternions and their composition prominent so I'm gonna go through not a unit not a problem I was like I wondered well let me see what happens if I put it on your attorney in it no problem I can scale the one to it to I can scale these two three and four for instance and just my representative vector they algorithm just did it automatically and it actually works so general attorneys ableton on Rodriguez Walton eros wisely Rodriguez in diseases had a better understanding the alter of rotations than Hamilton did so I thought maybe he could give me a clue on finding the formula here are pictures and some important parts have been translated by RSP you I have because I my French isn't so good so um all compositions and voltage axis must be placed in such a way that I'm carrying out the two indicated rotations around the supposed to the intersecting axes it comes back to the original position if therefore through each of the given axes one place is the plane that makes with with that one of these two axes an angle equal to half the rotation relative to this axis and the intersections to planes will be sought for the resulting accent access arriving by virtue of the first rotation wins position symmetric with respect to the way and the two exits arriving back and if you reverse the order in the rotation the axis comes never only any talks about that yet you tell us about so here a new let's just say what if you rotate this one forward by half the angle the same vector to a orthogonal vector in display and backwards an orthogonal vector in its plane take the eleven cross moment began to internal application they cannot take the cross it just pops out thanks to Hamilton is exactly what he said to do and it's it's inspired so here's this composition for Gloria wrote that exactly down there it is and he actually even I only noticed that just recently from others translation characteristic district moreover there is between these two sorts of composition and analogy the rectilinear triangle for translation the spherical triangle and the signs of the half rotations of the signs proportional the size of the area so we have to worry about the spherical triangle for rotation oh my gosh and then here's his access nothing and he also talked about the angle here's the the formula for the angle except I don't know if I trust anyone who thinks a full circle is 400 degrees anyway and so the same angle and he talks about the non commutativity says if you do it in this order that the axes are symmetrical across the line between the acts ET i to writes that down bury and supposedly and that's their axis symmetric so i'm running through that wasn't a damn scary it was in this journal with Koshi with blue shade with Koshi Delanie you been that's helped them single bank because apparently the same flamenco operator Catalan and Olivia and llama Dylan a so this was kind of not only that and just like Johannes discovered it can be wrote about him not only 1845 but it needs to create these are quote originals there's a there's an original I think I've had week other one we build sternum talked about blue tipz volume and stuff like that so there are here just both of Rodriguez and the white words this is Wyler's of matrix from axis and rotation and mrs. Rodriguez looseness which are fun here's white mark have the map from returning this and matrices there's there's the three there's the three by three components the maximum and so estimated miss no three so he did this part and run previous figured up the thing to make that a little more fizzle and we're to fade over to come is the last slide I think where did they come over to come from if you want if you will keep my wheels about theta times the axis and you might expand in it you get the rotation matrix now same context you know once you get to the ball of radius PI you exponentiate it at this time you fall off and this end then they're not periodic so it doesn't come back be prompt so what if you tried sine theta times the axis the problem is when theta is PI you lose all your access information and even when it doesn't you lose its to one so there are two things with the same song watch that you do notice there are two fun two values on the circle with the same side so all the way up how could you fix that let's cut this here and shrink it here and this will be the last thing they'll probably say is cut that bring it back to half but make that map theta over two suddenly this is one to one from the R sine theta over 2 sine theta over 2 times the access unit axis if you think of it as northern hemisphere is a one-to-one map is one to one with 3d rotations with us so three sine theta over two so three six in the unit quaternions in its vector pirate alone I mean you can basically solve for the scalar part of it to you in a quaternion square root of 1 minus sine squared plus or minus until that's where you got a double one and that's how Rodrigues ended up figuring out oh if I'm gonna make this continuous again but still periodic okay you have one minute but ask questions anybody have any questions for Bob any questions that was really really wonderful anyone have any questions for Bob okay I think we can put the light oh god there's a question over there something on one of your slides I think you know because we've correspondent about this but at one point a few slides back we talked about how one can think of a unit quaternion as this pair of unit vectors we're an equivalence relation equivalence relation you expressed it as same dotted cross product but that's equivalent to saying that the angle between them and the plane they lie there I mean they each pair determines is saying they're parallel then it can be any direction yeah and so there are what I thought have some fascinating and I didn't know this way of thinking about it until I read Hamilton so I've been in the business of reading Hamlet yeah that's two months as I think you know and so I read quite a lot of his first book his lectures on quaternions which is I don't know five or six hundred minutes anyway if you read the first several chapters you find out what his view of how he presented quaternions was and his it's completely unlike how we think of quaternion today he never he never gets to talking about even addition of quaternions or associativity of multiplication or even the quad the quadruple you know coordinate interpretation vectors but he but instead introduces quarter nians essentially as the we can think of it as introducing them as these ordered pairs of unit and and and his point of view there is that this is how you should think of a quarter knee as being a quotient roved of vectors rather than any other ways I'm not it fascinating that Tama quaternion and I suppose realize what form your name is a quotient of them yes it's fascinating that you and so is not different a cent difference in its owners via W inverse its present its quotient it's about how you get from one thing to another all right if you think about how you get from one thing to another how you get from this vector for this one you're essentially talking about a quotient so the forgiving is he introduced the tracer the difference if I hadn't seen that it was obvious I just blown away to find out that to him quaternions or quotients of that guy's formally is it's it's the W inverse whether that's addition inverse then its are two points is this equivalence class and you can formulate it instead of book formulating it as duck cross-product I can say these two are equivalent if and only if there's a rotation of space that takes this one and this one and this one this one just like a translation that takes this point to this point so you can and that's probably where he was going to them and this does to that spherical art interpretation that you try to talk about here much but which he then talks a lot about right but I was amazed to see that that Rodriguez actually really wrote spherical triangle heads like that so I think I should team up to translate nicely alright that we can you did write about and he wrote about spherical geometry too we mentioned about that yes let me know if you may continue to keep your 3d glasses because I can give you a wink or just woke up from a lady's floor mat yes it's a fun thing for great for teaching and outreach things - thank you