Quaternion units

Salomon, Y. and Avnir, D. (1999) Continuous symmetry measures Finding the closest C2-symmetric object or closest reflection-symmetric object using unit quaternions. J. Comput. Chem. 20, 772-780. 

Exercise 1.15 (Used in Section 4.1) Show that the product of two unit quaternions is a unit quaternion. (Hint Brute calculation will suffice, but the geometry ofSf may provide more insight think of the right-hand quaternion in the multiplication as a unit vector in R", think of the left-hand quaternion as a linear transformation of. )... 

A slightly more complicated example is the set of unit quaternions (defined in Section 1.5) with its usual multiplication. By Exercise 1.14, the multiplication is associative. The quaternion I e Q is the identity element. Also, for any unit quaternion u xi yj -+ zk we have -f 2 j... 

It follows from this calculation that m — xi — yj — zk is the inverse of u -fxi yj -I- zk. We are almost done proving that the unit quaternions form a group. (Any reader puzzled to find that we are not completely done should pause to think about what might be left to do.) Note that Definition 4.1 requires that the product of two elements of G should itself lie in G. We know that the product of any two quaternions is a quaternion, but to be complete we must show that the product of any two unit quaternions is a unit quaternion. See Exercise 1.15. 

There is a Lie group isomorphism between unit quaternions and the special unitary group 50 (2). Define a function T from the unit quaternions to 50 (2) by... 

Where z Owe have 1—zz —x —y 0, so the expression on the right-hand side is a differentiable function of zz, x and y. Hence 4z is differentiable at unit quaternions with z yt 0. A similar argument shows that P is differentiable at any unit quaternion with at least one nonzero coefficient. But each unit quaternion has at least one nonzero coefficient, so we have shown P to be differentiable on its domain. An almost identical argument shows that 4 is differentiable. Hence 4 is an isomorphism of Lie groups. 

Exercise 4.34 (SU (2) and the unit quaternions) Recall the functions f, fj and f)i,from Exercise 2.8. Show that the restrictions off, f and fk to the unit circle T are group homomorphisms whose range lies in the unit quaternions. Call their images T, Tj and Tk, respectively. Write the images ofTj, Tj azjJ Tk under the homomorphism explicitly a5 3 x 3 matrices. 

Next we show that this integral is invariant under group multiplication on the left. Recall from Section 4.2 that SU(2) is isomorphic to the unit quaternions. From Exercise 4,25 we know that multiplication of a unit quaternion q on the left by a unit quaternion qo corresponds to the product of a matrix in 5(9(4) (corresponding to qo) and a vector in 5- C (corresponding to q). See Figure 6.3. 

The above analysis shows that the set of turns To Tn E, E-, i = 1,2,3, provides a geometric realization of the quaternion group and thus establishes the connection between the quaternion units and turns through rc/2, and hence rotations through % (binary rotations). This suggests that the whole set of turns might provide a geometric realization of the set of unit quaternions. Section 12.5 will not only prove this to be the case, but will also provide us with the correct parameterization of a rotation. 

Theorem 1. Let a R3 be a unit vector and define the unit quaternion... 

Horn BKP. Closed-form solution of absolute orientation using unit quaternions. J Opt Soc Am A 1987 4(4) 629-42. 

We see that the imaginary unit i itself satisfies the first two criteria, but obviously not the last. The solution is that q is one of the quaternion units. Quaternions were developed by Hamilton as a further extension of the complex numbers in a progression ... 

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