Mannheim offsets of ruled surfaces under the 1-parameter motions
Resumo
In this study, the situation of Mannheim offsets of ruled surfaces under the 1-parameter motions is investigated. Firstly, relationships between geodesic Frenet trihedrons of Mannheim offsets of ruled surfaces are obtained and the relationship between the curvatures of the surface pairs is examined. Also, change of integral invariants the surface pairs under the 1-parameter motions is studied. Finally, the relevant example is given for every Mannheim offsets of ruled surfaces.
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Referências
K. Akdemir, M. Calıskan and N. Kuruoglu, On the pair of konoidal ruled surfaces under the homothetic motions of the Euclidean space En, Pure and Applied Mathematics Sciences, 4 (1997), 39-43.
J. Bertrand, Memoire sur la teorie des courbes a double courbure, ComptesRendus Des Seances De I'Academie Des Sciences, 31 (1850), 623-624.
M. P. Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, 1976.
L. Cui and J. S. Dai, From sliding-rolling loci to instantaneous kinematics: An adjoint approach, Mechanism and Machine Theory. 85 (2015), 161-171. https://doi.org/10.1016/j.mechmachtheory.2014.11.015
M. Calıskan and N. Kuruoglu, On the pair of konoidalaxoids, Pure and Applied Mathematica Sciences. 36 (1992), 7-13.
A. Gray, Modern differential geometry of curves and surfaces with mathematica, CRC Press ˙Inc., 1996.
C. Huyghens, Horologium oscillatoriumsive de motupendulorumadhorologiaaptato demonstrations geometricae, Paris; English translation: The pendulum clock or geometrical demontrations concerning the motion of pendula as applied to clocks, Iowa State University Press, 1673, available online at http://historical.library.cornell.edu/kmoddl/toc huygens1.html.
E. Kasap and N. Kuruoglu, On the some new characteristic properties of the pair of the Bertrand ruled surfaces, Pure Appl. Math. Sci. 53 (2001), 73-79.
E. Kasap, S. Yuce and N. Kuruoglu, The Involute-Evolute offsets of ruled surfaces, Trans.A. 33 (2009), 195-201.
Z. Lan, Z. Huijun and L. Liuming, Kinematic decomposition of coupler plane and the study on the formation and distribution of coupler curves, Mechanism and Machine Theory. 37 (2002), 115-126. https://doi.org/10.1016/S0094-114X(01)00054-4
M. Lipschutz, Schaum's outline of differential geometry, Schaum's Outlines, 1969.
H. Liu and F. Wang, Mannheim partner curves in 3-space, Journal of Geometry. 88 (2008), 120-126. https://doi.org/10.1007/s00022-007-1949-0
K. Orbay, and E. Kasap, On Mannheim partner curves in E3 , International Journal of Physical Sciences. 4 (2009), 261-264.
K. Orbay, E. Kasap, and ˙I. Aydemir, Mannheim offsets of ruled surfaces, Mathematical Problems in Engineering. 2009(160917) (2009), 1-9. https://doi.org/10.1155/2009/160917
B. Ravani, and T. S. Ku, Bertrand offsets of ruled and developable surfaces, Computer-Aided Design. 23 (1991), 145-152. https://doi.org/10.1016/0010-4485(91)90005-H
D. J. Struik, Lectures on classical differential geometry, Dover Publications, 1961.
D. L. Wang, J. Liu, and D. Z. Xiao, Kinematic differential geometry of a rigid body in spatial motion-I. A new adjoint approach and instantaneous properties of a point trajectory in spatial kinematics, Mechanism and Machine Theory. 32 (1997a), 419-432. https://doi.org/10.1016/S0094-114X(96)00075-4
D. L. Wang, J. Liu, and D. Z. Xiao, Kinematic differential geometry of a rigid body in spatial motion-II. A new adjoint approach and instantaneous properties of a line trajectory in spatial kinematics, Mechanism and Machine Theory. 32 (1997b), 433-444. https://doi.org/10.1016/S0094-114X(96)00076-6
D. L. Wang, J. Liu, and D. Z. Xiao, Kinematic differential geometry of a rigid body in spatial motion-III. Distribution of characteristic lines in the moving body in spatial motion, Mechanism and Machine Theory. 32 (1997c), 445-457. https://doi.org/10.1016/S0094-114X(96)00077-8
F. Wang and H. Liu, Mannheim partner curve in 3-Euclidean space, Mathematics in Practice and Theory. 37 (2007), 141-143.
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