Adaptive direct slicing of volumetric attribute data represented by trivariate B-spline functions

  • Yuhi Sasaki
  • Masahito Takezawa
  • Seungki Kim
  • Hiroshi Kawaharada
  • Takashi Maekawa


We introduce a framework for modeling of heterogeneous objects in terms of trivariate B-spline functions and a method for slicing them directly for additive manufacturing. We first fit volumetric attribute data associated with the geometry in terms of trivariate B-spline functions under the assumption that the geometric volume is already defined by the trivariate B-spline functions. Then, the B-spline volume and the associated attribute data are directly sliced without converting them to stereo-lithography format, resulting in a tool path with fewer errors. Furthermore, adaptive ray shooting is introduced in the slicing plane so that the zigzag tool path passes through all the tangential intersection points of the heterogeneous objects to represent all the feature points in the fabricated model. Complex examples illustrate the effectiveness of our method.


Additive manufacturing Direct slicing Trivariate B-spline function Heterogeneous object 


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  1. 1.
    (2012). Standard specification for additive manufacturing file format (AMF). ASTM standard F2915-12 1.1st editionGoogle Scholar
  2. 2.
    Böhm W, Farin G, Kahmann J (1984) A survey of curve and surface methods in cagd. Comput Aided Geom Des 1(1):1–60CrossRefzbMATHGoogle Scholar
  3. 3.
    Cho W, Maekawa T, Patrikalakis N (1996) Topologically reliable approximation of composite Bézier curves. Comput Aided Geom Des 13(6):497–520CrossRefzbMATHGoogle Scholar
  4. 4.
    Cormen TH, Leiserson CE, Rivest RL (1990) Introduction to Algorithms. Cambridge, Press, MAGoogle Scholar
  5. 5.
    Davis TA (2006) Direct methods for sparse linear systems. SIAM philadelphiaGoogle Scholar
  6. 6.
    Deng C, Lin H (2014) Progressive and iterative approximation for least squares b-spline curve and surface fitting. Comput Aided Des 47:32–44MathSciNetCrossRefGoogle Scholar
  7. 7.
    do Carmo PM (1976) Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood CliffszbMATHGoogle Scholar
  8. 8.
    Farouki RT, Rajan VT (1988) Algorithms for polynomials in Bernstein form. Comput Aided Geom Des 5 (1):1–26MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gao W, Zhang Y, Ramanujan D, Ramani K, Chen Y, Williams CB, Wang CCL, Shin YC, Zhang S, Zavattieri PD (2015) The status, challenges, and future of additive manufacturing in engineering. Comput Aided Des 69:65–89CrossRefGoogle Scholar
  10. 10.
    Huang P, Deng D, Chen Y (2013) Modeling and fabrication of heterogeneous three-dimensional objects based on additive manufacturing. In: ASME 2013 International Mechanical Engineering Congress and Exposition American Society of Mechanical EngineersGoogle Scholar
  11. 11.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39-41):4135–4195MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jamieson R, Hacker H (1995) Direct slicing of CAD models for rapid prototyping. Rapid Prototyp J 1 (2):4–12CrossRefGoogle Scholar
  13. 13.
    Kajiya JT (1982) Ray tracing parametric patches. Comput Graphics (SIGGRAPH ’82 Proceedings) 16 (3):245–254CrossRefGoogle Scholar
  14. 14.
    Khoda AKM, Ozbolat IT, Koc B (2013) Designing heterogeneous porous tissue scaffolds for additive manufacturing processes. Comput Aided Des 39(12):1507–1523CrossRefGoogle Scholar
  15. 15.
    Kineri Y, Wang M, Lin H, Maekawa T (2012) B-spline surface fitting by iterative geometric interpolation/approximation algorithms. Comput Aided Des 44(7):697–708CrossRefGoogle Scholar
  16. 16.
    Lin H (2010) The convergence of the geometric interpolation algorithm. Comput Aided Des 42(6):505–508CrossRefGoogle Scholar
  17. 17.
    Lin H, Jin S, Hu Q, Liu Z (2015) Constructing b-spline solids from tetrahedral meshes for isogeometric analysis. Comput Aided Geom Des 35-36:109–120MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lin H, Wang G, Dong C (2004) Constructing iterative non-uniform B-spline curve and surface to fit data points. Sci in China 47(3):315–331MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liu H, Maekawa T, Patrikalakis N, Sachs E, Cho W (2004) Methods for feature-based design of heterogeneous solids. Comput Aided Des 36(12):1141–1159CrossRefGoogle Scholar
  20. 20.
    Ma W, But WC, He P (2004) NURBS-Based adaptive slicing for efficient rapid prototyping. Comput Aided Des 36(13):1309–1325CrossRefGoogle Scholar
  21. 21.
    Maekawa T (1999) An overview of offset curves and surfaces. Comput Aided Des 31(3):165–173CrossRefzbMATHGoogle Scholar
  22. 22.
    Maekawa T, Matsumoto Y, Namiki K (2007) Interpolation by geometric algorithm. Comput Aided Des 39(4):313–323CrossRefGoogle Scholar
  23. 23.
    Martin T, Cohen E, Kirby R (2009) Volumetric parameterization and trivariate B-spline fitting using harmonic functions. Comput Aided Geom Des 26(6):648–664MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Martin T, Cohen E, Kirby R (2012) Direct isosurface visualization of hex-based high-order geometry and attribute representations. IEEE Trans Vis Comput Graph 18(5):753–766CrossRefGoogle Scholar
  25. 25.
    Martin W, Cohen E (2001) Representation and extraction of volumetric attributes using trivariate splines: a mathematical framework. In: Proceedings of the sixth ACM symposium on Solid modeling and applications, pp 234–240Google Scholar
  26. 26.
    Ozbolat IT, Koc B (2011) Multi-directional blending for heterogeneous objects. Comput Aided Des 43 (8):863–875CrossRefGoogle Scholar
  27. 27.
    Pan Y, Zhou C, Chen Y, Partanen J (2014) Multitool and multi-axis computer numerically controlled accumulation for fabricating conformal features on curved surfaces. ASME J Manufacturing Sci and Eng 136(3)Google Scholar
  28. 28.
    Park S, Lee K (1997) High-dimensional trivariate nurbs representation for analyzing and visualizing fluid flow data. Comput Graph 21(4):473–482CrossRefGoogle Scholar
  29. 29.
    Patrikalakis N, Maekawa T (2002) Shape interrogation for computer aided design and manufacturing. Springer-Verlag, HeidelbergGoogle Scholar
  30. 30.
    Paul R, Anand S (2015) A new steiner patch based file format for additive manufacturing processes. Comput Aided Des 63:86–100CrossRefGoogle Scholar
  31. 31.
    Piegl L, Tiller W (1997) The NURBS book 2nd Ed Springer-Verlag New York Inc.Google Scholar
  32. 32.
    Samanta K, Koc B (2005) Feature-based design and material blending for free-form heterogeneous object modeling. Comput Aided Des 37(3):287–305CrossRefGoogle Scholar
  33. 33.
    Schwahn ES (2015) Using controlled curing in a custom stereolithography-based 3D printing machine to obtain graded property variations. Master thesis, University of Nebraska- LincolnGoogle Scholar
  34. 34.
    Sethian JA (1999) Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geome- try, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University PressGoogle Scholar
  35. 35.
    Starly B, Lau A, Sun W, Lau W, Bradbury T (2005) Direct slicing of STEP based NURBS models for layered manufacturing. Comput Aided Des 37:387–397CrossRefGoogle Scholar
  36. 36.
    Wang X, Qian X (2014) An optimization approach for constructing trivariate B-spline solids. Comput Aided Des 46:179–191MathSciNetCrossRefGoogle Scholar
  37. 37.
    Yamaguchi F (1977) A method of designing free form surfaces by computer display (1st report)(in Japanese). Precision Machinery 43(2):168–173Google Scholar
  38. 38.
    Yang P, Qian X (2007) A B-spline-based approach to heterogeneous objects design and analysis. Comput Aided Des 39(2):95–111CrossRefGoogle Scholar
  39. 39.
    Yoshihara H, Yoshii T, Shibutani T, Maekawa T (2012) Topologically robust b-spline surface reconstruction from point clouds using level set methods and iterative geometric fitting algorithms. Comput Aided Geom Des 29:422–434MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2016

Authors and Affiliations

  • Yuhi Sasaki
    • 1
  • Masahito Takezawa
    • 1
  • Seungki Kim
    • 1
  • Hiroshi Kawaharada
    • 1
  • Takashi Maekawa
    • 1
  1. 1.Department of Mechanical EngineeringYokohama National UniversityYokohamaJapan

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