By Daniel Klawitter
After revising recognized representations of the gang of Euclidean displacements Daniel Klawitter offers a accomplished advent into Clifford algebras. The Clifford algebra calculus is used to build new versions that permit descriptions of the gang of projective modifications and inversions with admire to hyperquadrics. Afterwards, chain geometries over Clifford algebras and their subchain geometries are tested. the writer applies this concept and the built ways to the homogeneous Clifford algebra version akin to Euclidean geometry. furthermore, kinematic mappings for exact Cayley-Klein geometries are constructed. those mappings permit an outline of present kinematic mappings in a unifying framework.
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Additional resources for Clifford Algebras: Geometric Modelling and Chain Geometries with Application in Kinematics
15. For a k-dimensional linear subspace S spanned by [v1 , . . , vk ] we compute v1 ∧ v2 ∧ . . ik ei1 ∧ . . ∧ eek . ik are called Grassmann coordinates of S . Note that by Th. 1 Grassmann coordinates are deﬁned only up to a constant scalar factor. Therefore, Grassmann coordinates can be interpreted as homogeneous coordinates. , decomposable. Therefore, we need more relations to describe simple elements. Here we do not investigate these relations, the interested reader is referred to . To clarify these concepts, we give an example.
Dual Objects Planes and spheres are modelled by decomposable elements of grade four. Dualization means multiplication with the pseudoscalar. Therefore, the dual elements are of grade one. Since the dual element describes the orthogonal complement of the object, it is clear why we compute AI · ηx = 0, x ∈ R3 to get the point set described by NOG (A) = NIG (AI). Tangent Blades There are also linear subspaces in the homogeneous model that are tangent to the horosphere. It is clear that they cannot represent a point, but they should be interpreted as set of points, since they touch the horosphere in a speciﬁc null vector corresponding to a point.
There exists a three-dimensional generator space V 3 ⊂ S26 whose points do not correspond to displacements. If we denote a point P ∈ P7 (R) by P = (a0 , . . , a3 , c0 , . . , c3 )T R, this exceptional space V 3 is given by a0 = a1 = a2 = a3 = 0. Therefore, the point set S26 \V 3 is the image of the Euclidean displacements. Thus, the image space is a sliced quadric S26 \V 3 , a pseudo algebraic variety, so to say. Hence, we have a bijective mapping ζ : SE(E) → S26 \V 3 ⊆ P7 (R), SE(3) α → A = (a0 , a1 , a2 , a3 , c0 , c1 , c2 , c3 )T R.