ℝ-Mapping of Interacting Part Surfaces

  • Stephen P. RadzevichEmail author


A novel method of the surfaces mapping, namely ℝ-mapping of the interacting part surfaces is disclosed. The preliminary remarks on the developed approach is followed by an in-detail consideration of the concept underlying in the ℝ-mapping of the interacting part surfaces. Then, principal features of ℝ-mapping of a part surface P1 onto another part surface P2 are disclosed. Due to ℝ-mapping of surface returns an equation of the mapped surface in natural representation, namely, in terms of fundamental magnitudes of the first and of the second order, the derived equation of the mapped surface is required been reconstructed and been represented in a convenient reference system. This issue got a comprehensive discussion in this chapter of the monograph. Consideration in the chapter end with two examples of implementation of the discussed method of part surfaces mapping.


  1. 1.
    Shishkov, V. A. (1948). Elements of kinematics of generating and conjugating in gearing. In Theory and Computation of Gears (Vol. 6). Leningrad, LONITOMASH.Google Scholar
  2. 2.
    Shishkov, V. A. (1951). Generation of surfaces using continuously indexing method (p. 152). Moscow: Mashgiz.Google Scholar
  3. 3.
    Radzevich, S. P. (2010). Briefly about the kinematic method of surfaces generation and on history of equation of contact in the form n ∙ V = 0. Theory of Mechanisms and Machines 8(15), 42–51.
  4. 4.
    Radzevich, S. P. (1987). A method for designing of the optimal form-cutting-tool for machining of a given sculptured surface on multi-axis NC machine. Pat. No. 4242296/08 (USSR), Filed: March 31, 1987.Google Scholar
  5. 5.
    Radzevich, S. P. (1987). Profiling of the form cutting tools for sculptured surface machining on multi-axis NC machine. In Proceedings of the Conference: Advanced Designs of Cutting Tools for Agile Production and Robotic Complexes, Moscow, MDNTP (pp. 53–57).Google Scholar
  6. 6.
    Radzevich, S. P. (1989). Profiling of the form cutting tools for machining of sculptured surface on multi-axis NC machine. Stanki i Instrument, 7, 10–12.Google Scholar
  7. 7.
    Radzevich, S. P. (1991). Sculptured surface machining on multi-axis NC machine (Monograph) (192p). Kiev: Vishcha Schola.Google Scholar
  8. 8.
    Pat. No. 1449246 (USSR). A method of experimental simulation of machining of a sculptured surface on multi-axis NC machine. S.P.Radzevich. Filed: February 17, 1987, Int. Cl. B 23 C, 3/16.Google Scholar
  9. 9.
    Radzevich, S. P. (2007). A novel method for mathematical modeling of a form-cutting-tool of the optimum design. Applied Mathematical Modeling, 31, 2369–2654.CrossRefGoogle Scholar
  10. 10.
    Radzevich, S. P. (2001). Fundamentals of surface generation (Monograph) (592p). Kiev: Rastan.Google Scholar
  11. 11.
    Radzevich, S. P. (2010). Gear cutting tools: Fundamentals of design and computation (p. 786). Boca Raton, Florida: CRC Press.CrossRefGoogle Scholar
  12. 12.
    Radzevich, S. P. (2008). Kinematic geometry of surface machining (p. 536). Boca Raton, Florida: CRC Press.Google Scholar
  13. 13.
    Radzevich, S. P. (2004). Mathematical modeling of contact of two surfaces in the first order of tangency. Mathematical and Computer Modeling, 39(9–10), 1083–1112.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Radzevich, S. P. (2002). ℝ-mapping based method for designing of form cutting tool for sculptured surface machining. Mathematical and Computer Modeling, 36(7–8), 921–938.CrossRefGoogle Scholar
  15. 15.
    Radzevich, S. P. (2012). Theory of gearing: Kinematics, geometry, and synthesis (856p). Boca Raton, Florida.Google Scholar
  16. 16.
    Radzevich, S. P. (2018). Theory of gearing: Kinematics, geometry, and synthesis (2nd Ed., 898p). Boca Raton, Florida.Google Scholar
  17. 17.
    Radzevich, S. P. (1991). Differential-geometrical method of surface generation (300p) (Doctoral thesis), Tula, Tula Polytechnic Institute.Google Scholar
  18. 18.
    Struik, D. J. (1961). Lectures on classical differential geometry (2nd ed., p. 232). Massachusetts: Addison-Wesley Publishing Company Inc.zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Southfield Innovation CenterEaton CorporationSouthfieldUSA

Personalised recommendations