Quaternions:We will now delve into a subject, which at first may seem quite unrelated.Butkeep the above expression in mind, since it will reappear in most surprising way. This storybegins in the early 19th century, when the great mathematician William Rowan Hamiltonwas searching for a generalization of the complex number system.Imaginary numbers can be thought of as linear combinations of two basis elements, 1 andi,which satisfy the multiplication rules 12= 1,i2=-1 and 1·i=i·1 =i. (The interpretationofi=√-1 arises from the second rule.) A complex numbera+bican be thought of as avector in 2-dimensional space (a, b). Two important concepts with complex numbers are themodulus, which is defined to be√a2+b2, and theconjugate, which is defined to be (a,-b). Invector terms, the modulus is just the length of the vector and the conjugate is just a verticalreflection about thex-axis. If a complex number is of modulus 1, then it can be expressedas (cosθ,sinθ).Thus, there is a connection between complex numbers and 2-dimensionalrotations. Also, observe that, given such a unit modulus complex number, its conjugate is(cosθ,-sinθ) = (cos(-θ),sin(-θ)). Thus, taking the conjugate is something like negatingthe associated angle.Hamilton was wondering whether this idea could be extended to three dimensional space. Youmight reason that, to go from 2D to 3D, you need to replace the single imaginary quantityiwith two imaginary quantities, sayiandj.Unfortunately, this this idea does not work.After many failed attempts, Hamilton finally came up with the idea of, rather than using twoimaginaries, instead using three imaginariesi,j, andk, which behave as follows:i2=j2=k2=ijk=-1ij=k, jk=i, ki=j.Combining these, it follows thatji=-k,kj=-iandik=-j.The skew symmetry ofmultiplication (e.g.,ij=-ji) was actually a major leap, since multiplication systems up tothat time had been commutative.)Hamilton defined aquaternionto be a generalized complex number of the formq=q0+q1i+q2j+q3k.Thus, a quaternion can be viewed as a 4-dimensional vectorq= (q0, q1, q2, q3).The firstquantity is a scalar, and the last three define a 3-dimensional vector, and so it is a bit moreintuitive to express this asq= (s, u), wheres=q0is a scalar andu= (q1, q2, q3) is a vectorin 3-space. We can define the same concepts as we did with complex numbers:Conjugate: q*= (s,-u)Modulus:|q|=pq20+q21+q22+q23=ps2+ (u·u)Unit Quaternion: qis said to be a unit quaternion if|q|= 1Quaternion Multiplication:Consider two quaternionsq= (s, u) andp= (t, v):q=(s, u) =s+uxi+uyj+uzkp=(t, v) =t+vxi+vyj+vzk.Lecture 67Spring 2018
CMSC 425Dave Mount & Roger EastmanIf we multiply these two together, we’ll get lots of cross-product terms, such as (uxi)(vyj),but we can simplify these by using Hamilton’s rules. That is, (uxi)(vyj) =uxvh(ij) =uxvhk.
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Linear Algebra, Rotation, Complex number, Scaling, Quaternions and spatial rotation, affine transformation