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The study variety of conformal kinematics. (English) Zbl 1503.70004

The group of conformal displacements of \(\mathbb{R}^3\) is isomorphic to \(\mathrm{SO}(4,1)\) and generated by the Euclidean similarities and an inversion in a sphere. The authors construct a 10-dimensional projective variety \(\mathcal{S}\subset\mathbb{P}^{15}\) of degree 12 and a smooth projective hyperquadric \(\mathcal{N}\subset\mathbb{P}^{15}\) such that the Zariski open subset \(\mathcal{S}\setminus\mathcal{N}\) is isomorphic to the group of conformal displacements \(\mathrm{SO}(4,1)\). The authors call \(\mathcal{S}\) the Study variety of conformal kinematics and \(\mathcal{N}\) its null quadric. The projective space \(\mathbb{P}^{15}\) is identified with the projectivization of the vector space associated to the even subalgebra \(\operatorname{CGA}_+\) of the conformal geometric algebra of 3-dimensional Euclidean geometry. The subalgebra \(\operatorname{CGA}_+\) is represented in terms of a 4-dimensional algebra over the quaternions and the authors use this four-quaternion representation to describe the ideal of \(\mathcal{S}\) in terms of 10 explicit bilinear polynomials. Theorem 3 of the article states that the straight lines in \(\mathcal{S}\) containing the identity correspond to certain conformal motions as described in [L. Dorst, Math. Comput. Sci. 10, 97–113 (2016; Zbl 1341.65006)].
The Study variety of conformal kinematics \(\mathcal{S}\) can be seen as a generalization of the Study quadric, where the Study quadric \(\mathcal{Q}\subset\mathbb{P}^7\) and its null cone \(\mathcal{C}\subset\mathbb{P}^7\) are hyperquadrics such that \(\mathcal{Q}\setminus\mathcal{C}\) is isomorphic to the group of Euclidean displacements \(\operatorname{SE}(3)\). The article shows how to recover \(\mathcal{Q}\) from \(\mathcal{S}\) as a subvariety. In comparison, the straight lines in \(\mathcal{Q}\) containing the identity correspond to rotations around a fixed axis or translations in a fixed direction. They appear naturally in kinematics for the decomposition of complicated motions into simpler motions. Algebraically, such decompositions correspond to factorizations of poynomials with coefficients in the dual quaternions. The generalization of such factorizations to the conformal setting is one of the motivations for the article under consideration.

MSC:

70E45 Higher-dimensional generalizations in rigid body dynamics
15A66 Clifford algebras, spinors
15A67 Applications of Clifford algebras to physics, etc.
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
51B10 Möbius geometries
51F15 Reflection groups, reflection geometries

Citations:

Zbl 1341.65006
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References:

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