Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T01:00:19.060Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Geometry of Lineal Elements onA Sphere, Euclidean Kinematics, andElliptic Geometry

Published online by Cambridge University Press:  20 November 2018

J. M. Feld
J. M. Feld*
Affiliation:
Queens College Flushing, N. Y.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The geometry of slides and turns of oriented lineal elements in the plane was first studied by Kasner [10]. Slides and turns generate whirls, which constitute a three-parameter group W3. The product of W3 and M3, the three-parameter group of Euclidean displacements in the plane, yields a sixparameter group of whirl-motionsG6. The geometry of turbines, and also of general series of lineal elements, under G6 was investigated by Kasner in [10] and, in subsequent papers, by Kasner and DeCicco, particularly in [3], [4], [11], [12].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 93 - 110
Copyright
Copyright © Canadian Mathematical Society 1952

References

1. Blaschke, W., Euklidische Kinematik und nichteuklidische Geometrie, Z. Math. Phys., vol. 60 (1911), 61-91 and 203–204.Google Scholar
2. Blaschke, W., Ebene Kinematik (Leipzig and Berlin, 1938).Google Scholar
3. DeCicco, J., The geometry of whirl series, Trans. Amer. Math. Soc, vol. 43 (1938), 344–358.Google Scholar
4. DeCicco, J., The differential geometry of series of lineal elements, Trans. Amer. Math. Soc, vol. 46 (1939), 348–361.Google Scholar
5. Feld, J. M., The geometry of whirls and whirl-motions in space, Bull. Amer. Math. Soc, vol. 47 (1941), 927–933.Google Scholar
6. Feld, J. M., Whirl-similitudes, Euclidean kinematics, and non-Euclidean geometry, Bull. Amer. Math. Soc, vol. 48 (1942), 783–790.Google Scholar
7. Feld, J. M., On a representation in space of groups of circle and turbine transformations in the plane, Bull. Amer. Math. Soc, vol. 50 (1944), 930–934.Google Scholar
8. Feld, J. M., A kinematic characterization of series of lineal elements in the plane and of their differential invariants under the group of whirl-similitude s and some of its subgroups, Amer. J. Math., vol. 70 (1948), 129–138.Google Scholar
9. Grünwald, J., Ein Abbildungsprinzip, welches die ebene Geometrie und Kinematik mit der rdumlichen Geometrie verknüpft, S.B. Akad. Wiss. Wien., lia, vol. 80 (1911), 677–741.Google Scholar
10. Kasner, E., Th: group of turns and slides and the geometry of turbines,j, vol. 33 (1911), 193–202.Google Scholar
11. Kasner, E. and DeCicco, J., The geometry of turbines, flat fields and differential equationst, Amer. J. Math., vol. 59 (1937), 545–563.Google Scholar
12. Kasner, E., The geometry of the whirl-motion group G6: elementary invariants, Bull. Amer. Math. Soc, vol. 43 (1937), 399–403.Google Scholar
13. Narasinga Rao, A., Studies in turbine geometry I, J. Indian Math. Soc, vol. 3 (1938), 96-108; II, Proc Indian Acad. Sc, vol. 8A (1938), 179–186.Google Scholar
14. Scheffers, G., Isogonalkurven, Aequitangentialkurven und komplexe Zahlen, Math. Ann., vol. 60 (1905), 491–531.Google Scholar
15. Strubecker, K., Zur Geometrie sphärischer Kurvenscharen, Jber. dtsch. MatVer., vol. 44 (1934), 184–198.Google Scholar