M. Koecher’s research while affiliated with University of Münster and other places

... In particular, we will use that rotations can be parametrized by unit quaternions (which can be identified with the unit 3-sphere S 3 ), as well as the following formulae (see, e.g. [11,22]): ...

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Fast normalized cross-correlation for template matching with rotations

... The combinatorial and group theory properties of the number 6 are described in considerable detail in Shaw (1994), along with a tie-up with the properties of the Klein quadric in the space IPB 6 , B 6 = 2 V 4 . Other references include Janusz & Rotman (1982), Cameron & Van Lint (1991) and Tits (1991). ...

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Finite Geometry, Dirac Groups and the Table of Real Clifford Algebras

... To determine the index of a rotation R £ S0(3, Q), we use quaternions [21,22] and Cay ley's parametriza tion [22] with 4 coprime integers /c, A, p, v R (K ,\,P,V ) = Sx -I 2KV + 2An -2kv + 2\H 2/c/z + 2Aus -2/cA + 2pv \ -2kh + 2XU 2n\ + 2pv 8V ) ...

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Algebraic Solution of the Coincidence Problem in Two and Three Dimensions

... Graves communicated his discovery to Hamilton in a letter dated 4 January 1844 but it was only published in 1848 after having been rediscovered by Arthur Cayley (1821-1895) in 1845. Since then they have been called Cayley numbers; see Reference [153]. Wynn already discussed them in several of his publications [65,66,69,99]. ...

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The Legacy of Peter Wynn

... Over the real ground field, with being a scalar product, vector products were first considered, and classified, by Eckmann [5] in 1942, using topological methods. Other treatments are found in for example [9] and [8]. The technique of the present article is used in [3] for a comprehensive proof of the classification theorem in this special case. ...

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Vector product algebras

... by quaternions which were introduced by W.R. Hamilton in the middle of the 19th century after searching for more general number systems than complex numbers (e.g.[33] [34] which both (by definition) are real in case of real quaternions.The associative but not commutative multiplication law in the quaternion alge- ...

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Mathematical Aspects of SU (2) and SO (3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form

... Given any quadratic algebra A, Frobenius' lemma states that the set V = {v ∈ A | v 2 ∈ k1} \ (k1 \ {0}) of purely imaginary elements of A forms a linear subspace of A which is supplementary to k1 (cf. [9], [3], [11]). Accordingly, each x ∈ A has unique decomposition x = λ(x)1 + ι(x), with λ(x) ∈ k and ι(x) ∈ V . ...

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Quadratic division algebras revisited (Remarks on an article by J. M. Osborn)

... Hence, in particular every alternative division algebra is quadratic. In any quadratic algebra B, the subset ImB = {b ∈ B R1 | b 2 ∈ R1} ∪ {0} ⊂ B of purely imaginary elements is a linear subspace of B, and B = R1 ⊕ ImB (Frobenius [9]). We shall write α+v instead of α1+v when referring to elements in this decomposition. ...

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Normal forms for the G 2 -action on the real symmetric 7 × 7 -matrices by conjugation

... The basis element e 0 acts an identity and e 1 , e 2 , e 3 satisfy the following rules It is obvious that H is noncommutative. A well known fact about quaternions is any quaternion can be represented as 2 × 2 complex matrix through the bijective transformation [1,2]. In [3], a quaternion matrix which entries are quaternions have studied to a pair of complex matrices. ...

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Real and Hyperbolic Matrices of Split Semi Quaternions