Skip to main content
SpringerLink
Log in

Kinematic generation of ruled surface based on rational motion of point-line

  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

This paper studies representation of rigid combination of a directed line and a reference point on it (here referred to as a “point-line”) using dual quaternions. The geometric problem of rational ruled surface design is viewed as the kinematic problem of rational point-line motion design. By using the screw theory in kinematics, mappings from the spaces of lines and point-lines in Euclidean three-dimensional space into the hyperplanes in dual quaternion space are constructed, respectively. The problem of rational point-line motion design is then converted to that of projective Bézier or B-spline image curve design in hyperplane of dual quaternions. This kinematic method can unify the geometric design of ruled surfaces and tool path generation for five-axis numerical control (NC) machining.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ravani B, Wang J W. Computer aided geometric design of line constructs. J Mech Des, 1991, 113: 363–371

    Article  Google Scholar 

  2. Pottmann H, Lü W, Ravani B. Rational ruled surfaces and their offsets. Grap Mod Image Processing, 1996, 31(6): 544–552

    Article  Google Scholar 

  3. Peternell M, Pottmann H, Ravani B. On the computational geometry of ruled surfaces. Comput-Aided Des, 1999, 31: 17–32

    Article  MATH  Google Scholar 

  4. Ge Q J, Ravani B. Geometric Design of Rational Bézier line congruences and ruled surfaces using line geometry. In: Geometric Modelling. London: Springer-Verlag, 1998

    Google Scholar 

  5. Sprott K, Ravani B. Kinematic generation of ruled surfaces. Adv Comput Math, 2002, 17: 115–133

    Article  MATH  MathSciNet  Google Scholar 

  6. Zhu L M, Ding H, Xiong Y L. Third-order point contact approach for five-axis sculptured surface machining using non-ball-end tools (I): Third-order approximation of tool envelope surface. Sci China Tech Sci, 2010, 53: 1904–1912

    Article  MATH  Google Scholar 

  7. Zhu L M, Ding H, Xiong Y L. Third-order point contact approach for five-axis sculptured surface machining using non-ball-end tools (II): Tool positioning strategy. Sci China Tech Sci, 2010, 53: 2190–2197

    Article  MATH  Google Scholar 

  8. Ding H, Zhu L M. Global optimization of tool path for five-axis flank milling with a cylindrical cutter. Sci China Ser E-Tech Sci, 2009, 53: 2449–2459

    Article  Google Scholar 

  9. Jüttler B, Wagner M G. Computer-aided design with spatial rational B-spline motions. J Mech Des, 1996, 118(2): 193–201

    Article  Google Scholar 

  10. Ge Q J, Ravani B. Computer aided geometric design of motion interpolants. J Mech Des, 1994, 116(3): 756–762

    Article  Google Scholar 

  11. Park F C, Ravani B. Bézier curves on Riemannian manifolds and Lie Groups with kinematics applications. J Mech Des, 1995, 117: 36–40

    Article  Google Scholar 

  12. Suh S H, Chung D H, Lee B E, et al. Developing an integrated STEP-compliant CNC prototype. J Manuf Syst, 2002, 21(5): 350–362

    Article  Google Scholar 

  13. Zhang Y, Ting K L. On point-line geometry and displacement. Mech Mach Theory, 2004, 39(10): 1033–1050

    Article  MATH  MathSciNet  Google Scholar 

  14. Langeron J M, Duc E, Lartigue C. A new format for 5-axis tool path computation, using Bspline curves. Comput-Aided Des, 2004, 36: 1219–1229

    Article  Google Scholar 

  15. McCarthy J M. Introduction to Theoretical Kinematics. Cambridge: MIT Press, 1990

    Google Scholar 

  16. Roschel O. Rational motion design — a survey. Comput-Aided Des, 1998, 30(3): 169–178

    Article  Google Scholar 

  17. Bottema O, Roth B. Theoretical Kinematics. New York: Dover Publications, 1990

    MATH  Google Scholar 

  18. Zhang Y, Ting K L. On higher order point-line and the associated rigid body motions. J Mech Des, 2007, 129: 166–172

    Article  Google Scholar 

  19. Ge Q J. Projective convexity in computational kinematic geometry. In Proceedings of the ASME Design Engineering Technical Conference, Montreal, Canada, 2002. 715–723

  20. Ravani B, Roth B. Mappings of spatial kinematics. J Mech Trans Automatin Des, 1984, 106(3): 341–347

    Article  Google Scholar 

  21. Purwar A, Ge Q J. On the effect of dual weights in computer aided design of rational motions. J Mech Des, 2005, 127(5): 967–972

    Article  Google Scholar 

  22. Farin G. Curves and Surfaces for Computer Aided Geometric Design. 4th ed. New York: Academic Press, 1997

    MATH  Google Scholar 

  23. Xia J, Ge Q J. On the exact representation of the boundary surfaces of the swept volume of a cylinder undergoing rational Bézier and B-spline motions. J Mech Des, 2001, 123: 261–265

    Article  Google Scholar 

  24. Fleisig R V, Spence A D. A constant feed and reduced angular acceleration interpolation algorithm for multi-axis machining. Comput-Aided Des, 2001, 33(1): 1–15

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China

    XiaoMing Zhang, Han Ding & YouLun Xiong

  2. State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, 200240, China

    LiMin Zhu

Authors
  1. XiaoMing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. LiMin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Han Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. YouLun Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to LiMin Zhu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, X., Zhu, L., Ding, H. et al. Kinematic generation of ruled surface based on rational motion of point-line. Sci. China Technol. Sci. 55, 62–71 (2012). https://doi.org/10.1007/s11431-011-4609-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-011-4609-4

Keywords

Search

Navigation