Why Half Angle θ/2? in Spin or Quaternion Space? Intuitive Geometry

Transcript

hello I'm John in Beijing China this video is an introduction of my paper exact and intuitive geometry explanation why does a half angle rotation in Sphynx base or quaternion space correspond to a whole angle with rotation in normal 3d space why does a half angle rotation in catonian space or spin space correspond to a whole angle location in normal 3d space the question is equivalent to why a half angle in the great representation of as you to correspond to a whole angle in the representation of so3 usually we used computation of the abstract mathematics to answer a question and now I will give up as exact and intuitive geometry explanation on it in this paper in 3d space you see the reference frame x triple prime Y triple prep their triple prime here is a vector did the presume is length is 1 there is deep data angle Phi angle then there are true projection of that one is to text people prime Y triple prime this plain projection one protection another projection there through that triple prime there triple power the true projection the projection another projection another projection theta theta domain is PI granted when theta is the other are there to the triple prime is 1 when theta is pi there through the triple prime protection you- what does it joined XY form is follow that streaky reader body 3 the rigid body and a reference frame suppose through similar books there are common Oh there are any angles we will rotate one of the books to go to this state the reference frame is coincide step one you'll hit by our apples ed the book longest edge is X middle edge why the shortest edge said without hid the book fixed then tilt it there rotate about death by alpha about that for this state you'll hit by alpha about that to this did this did the middle edge the two middle-aged ill you know same vertical plane plane the Y prime axis is directly a path exactly a path alright pillow the Y 3 y 3 triple prime exists along the vertical the people from axis the record plan about all directed below step 2 you'll hit better about Y prime then advise this date to realize this did the true x axis is coinciding the true axis cool inside you'll hit data about Y prime Y prime fixed step 3 you'll help up gamma apart this X you'll see the data you'll hit the comma finish the work the X those that have true projections protect sections projection we call them in one complex plane complex plane this is the production of that that sweep that would read triple prime yeah they're two XY crippled friend this square plus this square is one there through the projection of that there people prying either positive or negative when he is negative we call it's the same direction we are there for this when the negative when this is negative we call it opposite direction this is always positive this positive or negative now we rotate this you'll take this described described wrote by their by their and first this is not the one line with this if it needs rotate our prime to this state this state this state they are on one line this state corresponds to this state Y prime is directly above or below the Y prime axis but the domain of the god rotate of the rotation the tow man is PI because this the rotation on another almost semi another semi plane if represent represent the negative the negative we must distinguish there to dead people prime poly oh well rotation about is dead then is definitely either positive or negative it's definite we must distinguish it's negative negative or positive positive negative this domain domain is for positive there to the triple prime projection this is for negative the domain of pi must correspond troupe I rotate about this so we must said alpha prime equals arfa please see one I'm Frank please see-through van prime to complex number we call it call them in the context plane on one line corresponds y prime directly above or below the Y triple prime axis generally it had has that angle alpha Prime Guillory generally you take alpha to this state corresponding this rotates by alpha prime is equals half this matrix represent this operation but this matrix the determinate of the matrix is not one although it launched for the representation of you two are not belongs to a skewed not special because this deployment need is not one we must modify modify [Applause] this matrix is determinate is why represent it's the representation of don't rotate about alpha what about you take rotation about then step 2 what is responding correspondence we use this we don't this this plane Devine theta angle we need to a cast that full protection through the same one these states this did the true complex number length is the same this who is found this we Bobby fired at cast this phase angle C 1 prime C 2 prime is this angle is the same negative gamma Prime then we are length is Connery is not the same we must adjust we use another another real plane another real plane there is a vector projection well your taste to this caught her over pi the true production the foot projection is equal your his bet ha trying better prime the domain of better prime is Pi is PI we must correspond this rotate by better about Y prime the better domain is true pi in 3d space this dogma or better prime is pi so we must take better prime is equal to half better this is better better you'll hit by better but Y prime to these stage another real plane we rotate this vector by better prime to this state to this state here is a matrix for this optician for this finally rotation about x axis we have already obtained a true matrix matrices in a complex plane to represent the 3d rotation about z axis and about y success Sibley they are now we can use the 2 matrix she sees of rotations to derive the third matrix matrix representing the 3d rotation about x axis the method comes from the finest lecture on physics of three rotate rotation by gamma about X it's equal to step a rotation by half pi about Y followed by step B your T by gamma about Z then followed by step C rotation by negative 1/2 PI about Y supply first step rotation about Y 1/2 pi 2 step rotation about dead gamma 2 step rotation by negative how has PI about what we get we obtain this matrix this is rotation this is correspond this corresponds to the 3d space to rotation about X x axis let's let's we step through step 1 step 2 step 3 step weather protection of their so that's the triple prime is negative 1 step 2 step 3 step these three steps correspond [Applause] who spawns to step 1 step 2 step 3 step [Applause] 1 step 2 step 3 what about quaternion Athenian athenaeus apiece have true projections into playing independent we thought that what we taught them c1c truth in one complex plane as the I Sigma Z Sigma that is a poly matrix when one of the poly matrices I Sigma Z is equivalent to the imaginary unit of I in quaternion as why you take rotation about Y representation I Sigma Y is equivalent to the imaginary unit J of the equation we call Tinian and rotation about X is mixes representation I Sigma I Sigma I Sigma X is equivalent to the imaginary unit K in a space of Catania thank you very much