Please cite this article in press as: J. Angulo, Geom I: Colour quaternions, J. Vis. Commun. (2009), d The definition of morphological operators for colour images requires a total ordering for colour points. A colour can be represented by different algebraic structures, in this paper we focus on real quaternions. The paper presents two main contributions. On the one hand, we have studied differ...

1 Quaternions and the three-dimensional sphere 3 1.1 Hamiltion’s quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Lie group S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 The normalizer of Q8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 The exponential map for S . . . . . . . . . . . . . . . . ....

In this paper, we propose a nonlinear adaptive position and attitude tracking controller for a rigid body that requires no information about the mass and inertia matrix of the body. Moreover, we provide sufficient conditions on the reference trajectory that guarantee mass and inertia matrix identification. The controller is shown to be almost globally asymptotically stable and can handle large ...

As we know, quaternions are very efficient for representing rotations with clear geometric meaning (rotation axis and angle) and only one redundancy. Unfortunately, they do not handle translations, which meanwhile can be made multiplicative along with rotations via the use of homogeneous coordinates. Despite also being 4-tuples, homogeneous coordinates are algebraically incompatible with quater...

Quaternions are well suited to describe and combine rotations in the usual 3D euclidian space. Therefore, an increasing interest has been shown especially by engineers for symbolically computing with quaternions. To give the ability to perform manipulations on symbolic expressions involving quaternions, we designed Quaterman, a Maple package and Quaternica, a Mathematica package. We present the...

The roots of -1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are studied and it is shown that there is an infinite number of non-trivial complexified quaternion roots (and two degenerate solutions which are the complex imaginary operator and the set of unit pure real quaternions). The non-trivial roots are shown t...

In this communication, introduce the split Mersenne and Mersenne-Lucas hybrid quaternions, also obtaining generating functions Binet formulas for these quaternions investigating some properties among them.