نتایج جستجو برای: real quaternions

تعداد نتایج: 530991  

Journal: :CoRR 2017
Cristina Flaut

In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved. Moreover, we give a set of invertible elements in split quaternion algebras and in split octonion algebras.

Journal: :CoRR 2018
Hannes Sommer Igor Gilitschenski Michael Bloesch Stephan Weiss Roland Siegwart Juan I. Nieto

Over the last decades quaternions have become a crucial and very successful tool for attitude representation in robotics and aerospace. However, there is a major problem that is continuously causing trouble in practice when it comes to exchanging formulas or implementations: there are two quaternion multiplications in common use, Hamilton’s multiplication and its flipped version, which is often...

2003
John H. Conway Derek A. Smith John C. Baez

Conway and Smith’s book is a wonderful introduction to the normed division algebras: the real numbers (R), the complex numbers (C), the quaternions (H), and the octonions (O). The first two are well-known to every mathematician. In contrast, the quaternions and especially the octonions are sadly neglected, so the authors rightly concentrate on these. They develop these number systems from scrat...

2003
John H. Conway Derek A. Smith John C. Baez

Conway and Smith’s book is a wonderful introduction to the normed division algebras: the real numbers (R), the complex numbers (C), the quaternions (H) and the octonions (O). The first two are well-known to every mathematician. In constrast, the quaternions and especially the octonions are sadly neglected, so the authors rightly concentrate on these. They develop these number systems from scrat...

2006
David R. Wilkins

5 Abstract Algebra 1 5.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 1 5.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 2 5.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 2 5.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.5 The General Associative Law . . . . . . . . . . . . . . . . . . 4 5.6 Identity elements...

2011
Casey Machen

The exponential map is important because it provides a map from the Lie algebra of a Lie group into the group itself. We focus on matrix groups over the quaternions and the exponential map from their Lie algebras into the groups. Since quaternionic multiplication is not commutative, the process of calculating the exponential of a matrix over the quaternions is more involved than the process of ...

2001
Harald P. Fritzer

A new and relatively simple version of the quaternion calculus is offered which is especially suitable for applications in molecular symmetry and structure. After introducing the real quaternion algebra and its classical matrix representation in the group SO(4) the relations with vectors in 3-space and the connection with the rotation group SO(3) through automorphism properties of the algebra a...

2012
David R. Wilkins

9 Vectors and Quaternions 51 9.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 51 9.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 52 9.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 53 9.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 54...

2010
David R. Wilkins

4 Vectors and Quaternions 47 4.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 49 4.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 50 4.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 51...

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