Quaternions 1.1 Quaternions are a class of hypercomplex numbers with four real components [1]. By analogy with the complex numbers being representable as a sum of real and imaginary parts (z  a + bi), quaternions can also be written as a linear combination: q  a + bi + cj + dk, (1) where 1, i, j, k make a group and satisfy the noncommutative rules: i2  j2  k2  -1, ij  ji  k, jk  -kj ...

an involution or anti-involution is a self-inverse linear mapping. in this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. moreover, properties and geometrical meanings of these matrices will be given as reflections in r^3.

Many quaternion numbers associated with Fibonacci and Lucas or even their generalizations have been defined widely discussed so far. In all the studies, coefficients of these quaternions selected from consecutive terms numbers. this study, we define other for usual quaternions. We also present some properties, including Binet's formulas d'Ocagne's identities, types quaternions.

Quaternions are an important tool to describe the orientation of a molecule. This paper considers the use of quaternions in matching two conformations of a molecule, in interpolating rotations, in performing statistics on orientational data, in the random sampling of rotations, and in establishing grids in orientation space. These examples show that many of the rotational problems that arise in...

Quaternions have long been integral to the field of computer graphics, due to their minimal and robust representation of rotations in three dimensional space. Dual quaternions represent a compact method of representing rigid body transformations (that is rotations and translations) with similar interpolation and combination properties. By comparing them to two other kinds of rigid transformatio...

Interpolating trajectories of points and geometric entities is an important problem for kinematics. To describe these trajectories, several algorithms have been proposed using matrices, quaternions, dual-quaternions, the Study quadric; last one allows embedding motors as 8D vectors into projective space P7, where interpolation rotations translations becomes a linear problem. Furthermore, confor...

Quaternions have been a popular tool in 3D computer graphics for more than 20 years. However, classical quaternions are restricted to the representation of rotations, whereas in graphical applications we typically work with rotation composed with translation (i.e., a rigid transformation). Dual quaternions represent rigid transformations in the same way as classical quaternions represent rotati...