Clifford Algebras: Geometric Modelling and Chain Geometries by Daniel Klawitter

By Daniel Klawitter

After revising recognized representations of the crowd of Euclidean displacements Daniel Klawitter supplies a accomplished creation into Clifford algebras. The Clifford algebra calculus is used to build new versions that permit descriptions of the crowd of projective adjustments and inversions with recognize to hyperquadrics. Afterwards, chain geometries over Clifford algebras and their subchain geometries are tested. the writer applies this thought and the built tips on how to the homogeneous Clifford algebra version comparable to Euclidean geometry. additionally, kinematic mappings for distinct Cayley-Klein geometries are built. those mappings let an outline of latest kinematic mappings in a unifying framework.

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A bundle of lines B and a field of lines F are mapped to a two-spaces Bμ and F μ that are contained entirely in M24 . with Ω(Li , Lj ) = 0, i, j = 1, 2, 3, i = j . These three lines possess image points spanning a two-space P12 (α : β : γ) = αL1 + βL2 + γL3 , α, β, γ ∈ R. The set of all points conjugate to this two-space is spanned by P12 ν = L1 ν∩L2 ν∩L3 ν . This defines the polar two-space of P12 . Since this polar two-space is the intersection of three tangent hyperplanes we conclude that the intersection P12 ν ∩ M24 corresponds to the set of all lines in L3 intersecting L1 , L2 , and L3 .

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. Study himself [61] gave a method to compute the Study coordinates.

K . Each swap of two elements causes a multiplication with −1. If we arrive at eii we can insert 1, or −1, or 0 as prescribed by the signature (p, q, r). ik , 0≤k≤n form the standard basis of the Clifford algebra. 10. An algebra element that is the product of invertible grade-1 elements is called a versor. ki , k1 < . . < ki form a basis of i V . Especially 0 V is the scalar part R and 1 V is isomorphic to the vector space V . Elements from 2 V are called bivectors and elements from n V are called pseudoscalars.

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