Quaternion Presentation

Transcript

good afternoon my name is Sylvester jallo today I'll be talk about Contin today my uh my discussion by continue is going to cover the functional in real world mathematical history operation and rotation so uh let's begin content is used for many aspect in life most of the time is 3D repres representation there the reflection rotation and scaling of the figure is can be uh rotated scaled and do using konum for formulas and H hit invention of Contin um important of Contin is basically for abstract algebra and H invented help a great deal to extend the understanding of algebra uh Contin uh Meuse with automorphism for to four for um dimensional space mostly in computer graphics and navigation system is used in map of control theory signal processing altitude control physics op opto mechanics and numerous other MTH Fields now we talk a little about Contin Ed for let's talk about the structure uh the structure of continia as you notice is kind of similar to complex number and they said complex number has a m imaginary and uh a real part uh continuing just the extension of that it has one imaginary but one real part I'm sorry and the imaginary the three imaginary has i j and k each one have a different axis rotation uh in a clockwise rotation uh the operation continue is just like it's just a complex number uh all the operations are the same except for modifcation which Ting a stu we talk about the one um continue each letter I J and K represent the of1 just like I complex number um Contin system are basic associative but non edium meaning non commun so if you take K * I will give you a different result than I * K because of the fact rotation going backwards uh each non zero cons MC inverse and it has also conjugate uh letter use represent is Q um sub a or H which basically hon the invented ofan uh and has a order of eight me it has eight elements plus or minus uh 1 plus - i plus- j plus - K let's start about the structure more about the structure okay like I was saying you before continu is not a communitive so you can take for example K * i j right so if you take you saw that the K axis and you rotate towards the rotate towards the I AIS go you go J but now if you go in the reverse order where you go I * K you get a negative J this is the top by rotation so you're going to reverse order so that's what um H got stuck at when he was trying to do his multiplication so it's possible to do multiplication to show a division but thing about it was the rotation you should multiply two letters you get another one but when you did the reverse order you are in epit okay that's the structure by now let's talk about the history about continue well let's start with we're going to start with Benjamin Rodriguez who was the first um person to start with with continue at the time was unknown uh Benjamin was born 1795 in border of France he was a French Banker a mathematician andal reform and sa did all kind of tricks in 1815 was water deor uh in mathematic at the University of Paris and when he did the destion the destion we call the now the Rodriguez formula is used while um in 1825 he published work on social reform banking which causes other math folk to be ignored because basic the math meditation other math people consider him not to be a mathematician because his work is basically on banking reforms and other stuff like that however in 184 he accomplished he published a what called transforming Group which led to the Discovery continue at the time it was but little one it was it was it was discovered prior to what William um Hamilton so let's talk about the inventor of the continue the inventor of Contin was William Warren Hamilton born August 4 1805 in Dublin Ireland where he spent rest of his life uh his father was always on the go he was raised by his uncle Reverend Hilton in 1822 he discovered an important error in the place um treaties and called sistic mechanics he cut the eye of the war astronomer at the time the war ason was the uh the professor of of astronomy at the College of Dublin in 1823 himton 10 in Dublin four years later he was aent professor of astronomy in astronom Ro Island so he took over the guy who discovered him who said he was a prodigy in 1827 he produced his first work three on system of RS over the theory of Optics November 183 he became obsessed after reading a a paper of a complex number and algebraic comp now his he was so obsessed that his gradual War consist considered complex number using the auto uh couet of a b which he he Co the word Vector which is not used throughout math to show the rotation describing the as compose so he used Bic uses two letters the A and the B of the imaginary parts to describe the compus and complete the modifcation addition and so forth he was hoping well he did that he was hoping to uh to develop threedimensional analogy of complex number however he was UN unsuccessful because he couldn't um multiply he was um in attempt to allowed the division he wasn't able to so he spent several years trying to figure that out but however in in o on October 16 1843 I walked on the Rock canal in Dublin he realized that it require three parts two imaginary Parts it require three three imaginary Parts inire one real number and two imaginary Parts um made a mistake right there uh he discover continue on October 16 1843 he spend the next 20 years trying to apply Contin to problems and I need map going to hold this time I'm trying to do with this however a study led to discover hton mechanics who was vied with field mechanics and electromagnetic however in 1865 he still living in Dublin he spread no's life W William W and himon died let the to commemorate his uh his trium describing Contin okay let's look after his after his death when Hamilton died he had several PR that Contin AOC his res his work uh in 1865 two people Petri and bement P Contin uh said that Contin can be used in physics and geometry to describe vectors for Hamilton coin uh entering through continia entirely to Contin however in 1880 Contin going to be displayed as a vector analysis which he developed Joshua Ward and Oliver hfield so these two people are also continue advocating that quing can be used to describe vectors analysis and M other stuff now let's talk about the rotation and container they use in map well container is using three dimensional and four dimensional but rotate stuff because it's very very important um the V consider axis of X Y um k j and I are consider axis of rotation represent the uh imaginary Parts Q is compet rotation about the unit vector by the angle so if you want to rotate any images you take it times Q by this angle okay so the vector for example this Vector x uh y J uh and z k i j and k represents the AIS the vector axis of rotation the vector can be rotated by angle through the linear operation uh as I give you right there with Q * the vector time the conjugate which Q is rotated by the angle cosine angle / 2 the vector sin / 2 um well when you Rota about the angle and taking the the conjugate of it it leaves the it leaves the vector length the same the direction is not effective so you can still rotate it but it leaves everything the same uh I give be more in depth about the formulas um uh this how the the rotation was was calculated at the bottom which is q v * the conjugate is can be written as where Q is SV and the conjug SV is basically just a minus sign to the um the complex part in Vector so you're just switching the sign Vector if you do the operation you just find out that um it's it's not altered at all what rotation do Ur uh since all contain are hyper sere multip by another it will the third continum meaning that it won't it will get you that will stay in the same group The Contin operation is interpr as the point of vector rotation with the respect to the a fixed coordinate frame um container represent two things in rotation the axis which been rotated and the angle which should been rotated by for through any con can form V of corner it rotation about the x axis of Q through an angle like I said before any AR conent can be related to to to the angle prob spetri for importantance emergency in rotation in relation to rotation is that falling spin represent Observer spinning three uh spical Direction the for Speed Matrix is just a 2x2 matrix which any qu can be written in so Bly any continum can be into this form you can just complete the rotation using this uh the operation uh falling mission is closely related because the um they um this some contining group if we can see like you can see on the next slide you can see that the PO spin Matrix which represents so I gave you three of them is is BIC um to the Quant in equation I give you the contining the Contin from the up which is H but if you notice the U is a d matri mean it's the real part and other part is just um complex Imes okay for can be expressed in the poly Matrix and which is WR into linear combination so that's how you can write into a line combination um that concludes my um discussion about the history of B rotation and using and relation to in life uh have three a full question for you uh what is the letter use to identify Contin and group what are the three letters the himton use uh what do the uh the square each represent what is the container use for how is polyeter related to container group and this conclude my topic by continue thank you for your time I appreciate you listening [Music] that