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Line-symmetric motions with respect to reguli. (English) Zbl 1337.70003

Summary: We investigate the set of axial reflections of the Euclidean 3-dimensional space with respect to the continuous set of generators of a regular or singular quadric of the projectively extended 3-dimensional space. These reflections define a continuous motion which is mapped, according to E. Study [Math. Ann. 39, 441–566 (1891; Zbl 02687628)], onto a conic section of the Study model of the set of all Euclidean displacements. This model is a hyperquadric in a real projective 7-dimensional space with a 3-dimensional exceptional generator space.
It will be shown that there is a bijection between the set of all conic sections of the Study hyperquadric and the set of motions defined by quadrics in the above mentioned way. Thus, on the one hand, a complete classification of conic sections with respect to the exceptional generator space is obtained as well as, on the other hand, the Euclidean type of the basic ruled quadric, to which the axial reflections are applied.

MSC:

70B15 Kinematics of mechanisms and robots
70B10 Kinematics of a rigid body
53A17 Differential geometric aspects in kinematics
51M30 Line geometries and their generalizations
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