Configurations harmoniques et quaternions

Les Rotations Et Les Quaternions

La construction du corps non commutatif H de quaternions est une variante plus compliquée de la construction du corps commutatif C de nombres complexes. Donc on commence en rappelant la construction de C à partir de R. Les nombres complexes s’écrivent a+ bi avec a, b ∈ R. Donc C est un espace vectoriel sur R de dimension 2 avec base {1, i}. La loi d’addition de C est celle que C possède en tant...

Understanding quaternions

The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared for its importance with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science.-Clerk Maxwell, 1869. Quaternions came from Hamil...

Generalized Quaternions

The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomorphism). The other one, D4, can be constructed as a semi-direct product: D4 ∼= Aff(Z/(4)) ∼= Z/(4) o (Z/(4))× ∼= Z/(4) o Z/(2), where the elements of Z/(2) act on Z/(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We wil...

Quaternions et Algèbres Géométriques, de nouveaux outils pour les images numériques couleur. (Quaternions and Geometric Algebras, new tools for digital colour images)

The main subject of this PhD thesis is colour image processing. The first methods dealing with these images consisted in applying existing greyscale processing alorithms on each of the three colour components. Colour processing has improved using perceptual colour spaces but also by considering colours as vectors. In this work, we follow the idea of colour modelization and we propose to encode ...

ON QUATERNIONS By