Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Contour parallel milling tool path generation for arbitrary pocket shape using a fast marching method


Contour parallel tool paths are among the most widely used tool paths for planer milling operations. A number of exact as well as approximate methods are available for offsetting a closed boundary in order to generate a contour parallel tool path; however, the applicability of various offsetting methods is restricted because of limitations in dealing with pocket geometry with and without islands, the high computational costs, and numerical errors. Generation of cusps, segmentation of rarefied corners, and self-intersection during the offsetting operations and finding a unique offsetting solution for pocket with islands are among the associated problems in contour tool path generation. Most of methods are inherently incapable of dealing with such problems and use complex computational routines to identify and rectify these problems. Also, these rectifying techniques are heavily dependent on the type of geometry, and hence, the application of these techniques for arbitrary boundary conditions is limited and prone to errors. In this paper, a new mathematical method for generation of contour parallel tool paths is proposed which is inherently capable of dealing with the aforementioned problems. The method is based on a boundary value formulation of the offsetting problem and a fast marching method based solution for tool path generation. This method handles the topological changes during offsetting naturally and deals with the generation of discontinuities in the slopes by including an “entropy condition” in its numerical implementation. The appropriate modifications are carried out to achieve higher accuracy for milling operations. A number of examples are presented, and computational issues are discussed for tool path generation.

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


  1. 1.

    Kramer TR (1992) Pocket milling with tool engagement detection. J Manuf Syst 11(2):114–123

  2. 2.

    Held M (2001) VRONI: an engineering approach to the reliable and efficient computation of Voronoi diagrams of points and line segments. Comp Geom Theor Appl 18(2):95–123

  3. 3.

    El-Midany TT, Elkeran A, Tawfik H (2006) Toolpath pattern comparison contour-parallel with direction-parallel. Proceedings of the Conference on Geometric Modeling and Imaging New Trends, 5–6 July, London, UK, p 77–82

  4. 4.

    Farouki RT (1985) Exact offset procedures for simple solids. Comput Aided Geom Des 2(4):257–279

  5. 5.

    Farouki RT (1992) Pythagorean-hodograph curves in practical use. Geometry Processing for Design and Manufacturing. SIAM Publishers, Philadelphia, pp 3–33

  6. 6.

    Persson H (1978) NC machining of arbitrary shaped pockets. Comput Aided Des 10(3):169–174

  7. 7.

    Klass R (1983) An offset spline approximation for plane cubic splines. Comput Aided Des 15(5):297–299

  8. 8.

    Tiller W, Hanson EG (1984) Offsets of two-dimensional profiles. IEEE Comput Graph Appl 4(9):36–46

  9. 9.

    Hoschek J (1988) Spline approximation of offset curves. Comput Aided Geom Des 5(1):33–40

  10. 10.

    Elber G, Lee IK, Kim MS (1997) Comparing offset curve approximation methods. IEEE Comput Graph Appl 17(3):62–71

  11. 11.

    Hansen A, Arbab F (1992) Algorithm for generating NC tool paths for arbitrarily shaped pockets with islands. ACM Trans Graph 11(2):152–182

  12. 12.

    Veeramani D, Gau YS (2000) Cutter-path generation using multiple cutting-tool sizes for 2-1/2D pocket machining. IIE Trans (Institute of Industrial Engineers) 32(7):661–675

  13. 13.

    Stori JA, Wright PK (2000) Constant engagement tool path generation for convex geometries. J Manuf Syst 19(3):172–183

  14. 14.

    Choi BK, Kim BH (1997) Die-cavity pocketing via cutting simulation. CAD Comput Aided Des 29(12):837–846

  15. 15.

    Saeed SEO, de Pennington A, Dodsworth JR (1988) Offsetting in geometric modelling. Comput Aided Des 20(2):67–74

  16. 16.

    Molina-Carmona R, Jimeno A, Rizo-Aldeguer R (2007) Morphological offset computing for contour pocketing. J Manuf Sci Eng, Transactions of the ASME 129(2):400–406

  17. 17.

    Bieterman MB, Sandstrom DR (2003) A curvilinear tool-path method for pocket machining. J Manuf Sci Eng, Transactions of the ASME 125(4):709–715

  18. 18.

    Chuang JJ, Yang DCH (2007) A Laplace-based spiral contouring method for general pocket machining. Int J Adv Manuf Technol 34(7–8):714–723

  19. 19.

    Kimmel R, Bruckstein AM (1993) Shape offsets via level sets. Comput Aided Des 25(3):154–162

  20. 20.

    Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. Proc Natl Acad Sci USA 93(4):1591–1595

  21. 21.

    Hassouna MS, Farag AA (2007) Multistencils fast marching methods: a highly accurate solution to the Eikonal equation on Cartesian domains. IEEE Trans Pattern Anal Mach Intell 29(9):1563–1574

  22. 22.

    Lorensen WE, Cline HE (1987) Marching cubes: a high resolution 3D surface construction algorithm. Comput Graph (ACM) 21(4):163–169

Download references

Author information

Correspondence to Sandeep Dhanik.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dhanik, S., Xirouchakis, P. Contour parallel milling tool path generation for arbitrary pocket shape using a fast marching method. Int J Adv Manuf Technol 50, 1101–1111 (2010).

Download citation


  • Contour parallel
  • 2.5D milling
  • Pocketing