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The collection of all quaternions is denoted byH and is called the real quaternionic algebra. This algebra was first introduced by Hamilton in 1843 (see [5, 6]), and is often called the Hamilton quaternionic algebra. It is well known thatH is an associative division algebra over R. For any a= a0 + a1i+ a2 j + a3k ∈H, the conjugate of a = a0 + a1i + a2 j + a3k is defined to be a = a0 − a1i− a2 j...

We discuss optimal rotation estimation from two sets of 3-D points in the presence of anisotropic and inhomogeneous noise. We rst present a theoretical accuracy bound and then give a method that attains that bound, which can be viewed as describing the reliability of the solution. We also show that an e cient computational scheme can be obtained by using quaternions and applying renormalization...

1.1. The matrix Ξkk′ In the main text, we are given two unit vectors μ1k and μ2k′ in R. We define Ξkk′ = Ξ(μ1k, μ2k′), where Ξ(u, v) ∈ R4×4 is defined by u (q ◦ v) = qΞ(u, v)q, where u = (ui, uj , uk), v = (vi, vj , vk), and q = (qi, qj , qk, qr). By standard quaternion rotation formula, we have u (q ◦ v) = ui uj uk T 1− 2q j − 2q k 2(qiqj − qkqr) 2(qiqk + qjqr) 2(qiqj + qkqr) 1− 2q i − 2...

Rigid body dynamics on the rotation group have typically been represented in terms of rotation matrices, unit quaternions, or local coordinates, such as Euler angles. Due to the coordinate singularities associated with local coordinate charts, it is common in engineering applications to adopt the unit quaternion representation, and the numerical simulations typically impose the unit length cond...

Structured real canonical forms for matrices in Rn×n that are symmetric or skewsymmetric about the anti-diagonal as well as the main diagonal are presented, and Jacobi algorithms for solving the complete eigenproblem for three of these four classes of matrices are developed. Based on the direct solution of 4 × 4 subproblems constructed via quaternions, the algorithms calculate structured orthog...

Let Γn = (γij)n×n be a random matrix with the Haar probability measure on the orthogonal group O(n), the unitary group U(n) or the symplectic group Sp(n). Given 1 ≤ m < n, a probability inequality for a distance between (γij)n×m and some mn independent F -valued normal random variables is obtained, where F = R, C or H (the set of real quaternions). The result is universal for the three cases. I...

The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real and complex numbers which have achieved universal acceptance. Serious attempts have been made at further extensions, such as Hamiltons quaternions, Grassmann’s...

Iztok Fister and Iztok Fister Jr. Abstract This paper introduces a novel idea for representation of individuals using quaternions in swarm intelligence and evolutionary algorithms. Quaternions are a number system, which extends complex numbers. They are successfully applied to problems of theoretical physics and to those areas needing fast rotation calculations. We propose the application of qu...

Quaternions are ordered quadruples of four numbers subject to specified rules of addition and multiplication, which can represent points in four-dimensional (4D) space and which form finite groups under multiplication isomorphic to polyhedral groups. Projection of the 8 quaternions of the dihedral group D2h, with only two-fold symmetry, into 3D space provides a basis for crystal lattices up to ...