×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wunderlich, W.: Kubische zwangläufe, Sitzungsberichte der akad. D. wiss. Wien mathem.-naturw. Klasse 193, 45-68 (1984) · Zbl 0555.53006
[2] Röschel, O.: Rationale räumliche zwangläufe vierter ordnung, Sitzungsberichte der akad. D. wiss. Wien mathem.-naturw. Klasse 194, 185-202 (1985) · Zbl 0594.53009
[3] Jüttler, B.: Über zwangläufige rationale bewegungsvorgänge, Sitzungsberichte der akad. D. wiss. Wien mathem.-naturw. Klasse 202, 117-132 (1993) · Zbl 0806.53011
[4] Mccarthy, J. M.: Geometric design of linkages, (2000) · Zbl 0955.70001
[5] Brunnthaler, K.; Schröcker, H. -P.; Husty, M.: A new method for the synthesis of bennett mechanisms, (2005)
[6] Pottmann, H.; Wallner, J.: Computational line geometry, (2001) · Zbl 1006.51015
[7] Röschel, O.: Rational motion design – a survey, Computer-aided design 30, No. 3, 169-178 (1998) · Zbl 0906.68175 · doi:10.1016/S0010-4485(97)00056-0
[8] Husty, M.; Karger, A.; Sachs, H.; Steinhilper, W.: Kinematik und robotik, (1997) · Zbl 0877.70001
[9] Selig, J. M.: Geometric fundamentals of robotics, (2005) · Zbl 1062.93002
[10] Giering, O.: Vorlesungen über höhere geometrie, (1982) · Zbl 0493.51001
[11] Pfurner, M.: Analysis of spatial serial manipulators using kinematic mapping. Doctoral Thesis, 2006.
[12] Study, E.: Von den bewegungen und umlegungen, Math. annal. 39 (1891) · JFM 23.0527.01
[13] Bottema, O.; Roth, B.: Theoretical kinematics, (1990) · Zbl 0747.70001
[14] Krames, J.: Über fußpunktkurven von regelflächen und eine besondere klasse von raumbewegungen, Monatshefte für Mathematik und physik 45, 394-406 (1937) · Zbl 0016.36705 · doi:10.1007/BF01708003
[15] Krames, J.: Zur kubischen kreisbewegung des raumes, Sitzungsberichte der akad. D. wiss. Wien mathem.-naturw. Klasse 146, 145-158 (1937) · Zbl 0017.22001
[16] Krames, J.: Zur geometrie des bennett’schen mechanismus, Sitzungsberichte der akad. D. wiss. Wien mathem.-naturw. Klasse 146, 159-173 (1937) · Zbl 0017.22002
[17] Harris, J.: Algebraic geometry: A first course, (1992) · Zbl 0779.14001
[18] Farin, G. E.: Curves and surfaces for computer aided geometric design: A practical guide, (2001) · Zbl 0694.68004
[19] Keller, O. -H.: Vorlesungen über algebraische geometrie, (1974) · Zbl 0292.14001
[20] Müller, E.; Krames, J.: Vorlesungen über darstellende geometrie, Konstruktive behandlung der regelflächen 3 (1931)
[21] Grünwald, A.: Die kubische kreisbewegung eines starren körpers, Z. math. Phys. 55 (1907) · JFM 38.0702.04
[22] Brauner, H.: Vorlesungen über Liniengeometrie. Unpublished script, 1972.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.