Kinematic mappings for Cayley–Klein geometries via Clifford algebras

Abstract

Abstract. This paper unifies the concept of kinematic mappings by using geometric algebras. We present a method for constructing kinematic mappings for certain Cayley-Klein geometries. These geometries are described in an algebraic setting by the homogeneous Clifford algebra model. Displacements correspond to Spin group elements. After that Spin group elements are mapped to a kinematic image space. Especially for the group of planar Euclidean displacements SE(2) the result is the kinematic mapping of Blaschke and Grünwald. For the group of spatial Euclidean displacements SE(3) the result is Study's mapping. Furthermore, we classify kinematic mappings for Cayley-Klein spaces of dimension 2 and 3.

Projective Metric Geometry and Clifford Algebras

“…The latter set can be made into a group in a natural way and as such it acts on the initial projective metric space P(V , Q). In Theorems 5.4, 5.5 and 5.6 we carry out a detailed study of this group action and its kernel, thereby extending previous work of Gunn [19,20], Jurk [32], Klawitter and Hagemann [36], Klawitter [35], Schröder [49] and others. Since the details are somewhat involved, an alternative point of view is adopted in Tables 1, 2 and 3.…”

“…Under the restrictions of Table 3 we still have a kind of "kinematic mapping", but here one element of PO (V , Q) is represented by an unordered pair of points from G(V Q). Some of the examples in [35, 3.4] and [36,Sect. 6] fit into the above concepts.…”

Projective metric geometry and Clifford algebras

“…The latter set can be made into a group in a natural way and as such it acts on the initial projective metric space P(V, Q). In Theorems 5.4, 5.5 and 5.6 we carry out a detailed study of this group action and its kernel, thereby extending previous work of C. Gunn [19], [20], R. Jurk [32], M. Hagemann and D. Klawitter [36], [35], E. M. Schröder [49] and others. Since the details are somewhat involved, an alternative point of view is adopted in Tables 1-3.…”

“…(c) The twisted adjoint representations of the quotient groups Lip × (V, Q)/F × and Lip × ( Ṽ, Q)/F × are equivalent by virtue of the isomorphism (36) and the given similarity ψ : V → Ṽ.…”

Points in the plane, lines in space

“…The line, τ x = τ y = ∆ = 0, doesn't meet the quadric of rigid-body displacements -unless we change the ground field to C. See [1,Ch. 11] and also [7] for more details.…”