Bir Uzay Eğrisi ve Regle Yüzey Arasında Bishop Çatısına Dayalı Bitişik Yaklaşımı

Abstract

Bitişik yaklaşım genellikle krank-rocker bağlantılarının kuplör eğrilerini ve uzay hareketinde katı cisimlerin geometrisini incelemek için kullanılır. Bu makalede, Bishop çatısına dayalı bir uzay eğrisi ve bir regle yüzey arasındaki bitişik yaklaşım sunulmaktadır. Ayrıca, Tip-1 ve Tip-2 Bishop çatılarının bileşenleri kullanılarak bir regle yüzey ifade edilmiştir. Ayrıca, bir regle yüzeye bitişik bir eğri için Bishop çatısına bağlı sabit nokta koşulları belirlenmiştir. Son olarak, regle yüzey ile onun bitişik eğrisi arasındaki ilişkiyi göstermek için dört örnek sunulmuştur.

Keywords

• Tip-1 Bishop çatısı
• Tip-2 Bishop çatısı
• Serret-Frenet çatısı
• bitişik eğri
• regle yüzey

Adjoint Approach Between A Spatial Curve and A Ruled Surface Based On The Bishop Frame

Abstract

The adjoint approach is usually used to study the crank-rocker linkages’ coupler curves and the geometry of rigid objects in spatial motion. In this paper, the adjoint approach between a spatial curve and a ruled surface based on the Bishop frame is presented. Also, a ruled surface by using the components of the Type-1 and Type-2 Bishop frames is expressed. Moreover, for a curve that adjoint to a ruled surface the fixed point conditions concerning the Bishop frame are determined. Finally, we presented four examples to show the relationship between the ruled surface and its adjoint curve.

Keywords

• Type-1 Bishop frame
• Type-2 Bishop frame
• Serret-Frenet frame
• adjoint curve
• ruled surface

References

1. Bishop, L.R. (1975). There is a more than one way to frame a curve, Amer. Math. Monthly. Vol 82, Issue 3, 246-251.
2. Hanson, A. J. and Ma, H.H. (1995). Parallel Transport Approach to Curve Framing, Tech. Math. Rep. 425, Indiana University Computer Science Department.
3. Gray, A. (1996). Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, Inc.
4. Lipschutz, M. (1969). Schaum's Outline of Differential Geometry, Schaum's Outlines.
5. Lucas, P., Ortega-Yagües, J.A. (2012). Bertrand curves in the three-dimensional sphere, Journal of Geometry and Physics, 62, 1903-1914.
6. Sasaki, S. (1955). Differential Geometry (in Japanese), Kyolitsu Press.
7. Wang, D. and Xiao, D.Z. (1993). Distribution of coupler curves for crank-rocker linkages, Mechanism and Machine Theory, 28, 671-684.
8. Wang, D., Liu, J. and Xiao, D.Z. (1997). 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, 419-432.

9. Wang, D., Liu, J. and Xiao, D.Z. (1997). 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, 445-457.
10. Goetz, A. (1970). Introduction to differential geometry, Addison-Wesley series in mathematics, Intermediate Mathematics Series; Addison Wesley Pub. Co.
11. Wang, D. and Wang, W. (2015). Kinematic Differential Geometry and Saddle Synthesis of Linkages, Wiley, 1 edition.
12. Bulut, V. (2018). A Spatial Curve Adjoining Another Spatial Curve Based on Bishop Frame, International Conference on Mathematical Advances and Applications (ICOMAA2018), Istanbul.