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Unconstrained and constrained curve fitting for reverse engineering

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Abstract

Curve fitting is commonly used in reverse engineering for the reconstruction of curves from measured points, and it is critically important to provide various kinds of curve-fitting algorithms to acquire curves that satisfy different constraint conditions. We divide the curve-fitting problem into unconstrained and constrained types. For the unconstrained type, three curve-fitting algorithms are investigated: general, smooth and extended curve fitting. The general curve fitting considers only the accuracy of the fitted curve; the smooth curve fitting can control both the accuracy and the fairness of the fitted curve, while the extended curve fitting can acquire a curve longer than the range of the measured points. For the constrained type, we propose three curve-fitting conditions: fixed end-points, closed curve and continuity to adjacent curves. Detailed discussion for each of the above cases is presented. Associated examples are also provided to illustrate the feasibility of the proposed algorithms.

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References

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

    Article  MATH  Google Scholar 

  2. Rogers DF, Fog NG (1988) Constrained B-spline curve and surface fitting. Comput Aided Des 21(10):641–648

    Article  Google Scholar 

  3. Sarkar B, Menq CH (1991) Parameter optimization in approximating curves and surfaces to measurement data. Comput Aided Geom Des 8:267–290

    Article  MATH  Google Scholar 

  4. Wang W, Pottmann H, Liu Y (2004) Fitting B-spline curves to point clouds by squared distance minimization. HKU CS Tech Report TR-2004-11

  5. Heidrich W, Bartels R, Labahn G (1997) Fitting uncertain data with NURBS. Proc. of Curves and Surfaces, Chamonix, France, pp 1–8

  6. Ma W, Kruth JP (1995) Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces. Comput Aided Des 27(9):663–675

    Article  MATH  Google Scholar 

  7. Fang L, Gossard DC (1995) Multidimensional curve fitting to unorganized data points by nonlinear minimization. Comput Aided Des 27(1):48–58

    Article  MATH  Google Scholar 

  8. Aszodi B, Czuczor S, Szirmay-Kalos L (2004) NURBS fairing by knot vector optimization. J WSCG 12(1–3)

    Google Scholar 

  9. Li W, Xu S, Zheng J, Zhao G (2004) Target curvature driven fairing algorithm for planar cubic B-spline curves. Comput Aided Geom Des 21:499–513

    MATH  Google Scholar 

  10. Poliakoff JF, Wong YK, Thomas PD (1998) An automated curve fairing algorithm for cubic B-spline curves. J Comput Appl Math 102:73–85

    Article  Google Scholar 

  11. Eck M, Hadenfeld J (1995) Local energy fairing of B-spline curves. Comput Suppl 10:129–147

    Google Scholar 

  12. Farin G, Sapidis N (1989) Curvature and fairness of curves and surfaces. IEEE Comput Graph Appl 9(2):52–57

    Article  Google Scholar 

  13. Szobonya L, Renner G (2002) Construction of curves and surfaces based on point clouds. Proc First Hungarian Conference on Computer Graphics and Geometry, Budapest, pp 57–62

  14. Yang H, Wang W, Sun J (2004) Control point adjustment for B-spline curve approximation. Comput Aided Des 36:639–652

    Article  Google Scholar 

  15. Park H, Kim K, Lee SC (2000) A method for approximate NURBS compatibility based on multiple curve refitting. Comput Aided Des 32:237–252

    Article  Google Scholar 

  16. Alhanaty M, Bercovier M (2001) Curve and surface fitting and design by optimal control methods. Comput Aided Des 33:167–182

    Article  Google Scholar 

  17. Benko P, Kos G, Varady T, Andor L, Martin R (2002) Constrained fitting in reverse engineering. Comput Aided Geom Des 19:173–205

    Article  Google Scholar 

  18. Pottmann H, Leopoldseder, Hofer M (2002) Approximation with active B-spline curves and surfaces. Proc of the Pacific Graphics IEEE Press 8–25

  19. Piegl L, Tiller W (1997) The NURBS book. Springer, Berlin Heidelberg New York

    Google Scholar 

  20. Leon SJ (1994) Linear algebra with applications, 4th edn. Prentice Hall, New York

    Google Scholar 

  21. Reklaitis GV, Ravindran A, Ragsdell KM (1984) Engineering optimization method and applications. Wiley

  22. Faux ID, Pratt MJ (1979) Computational geometry for design and manufacture. Ellis Horwood

  23. Lai JY, Ueng WD (2000) G2 continuity for multiple surfaces fitting. Int J Adv Manuf Technol 17(8):575–585

    Article  Google Scholar 

  24. Rogers DF, Adams JA (1990) Mathematical elements for computer graphics. McGraw-Hill

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Authors and Affiliations

  1. Mechanical Engineering Department, Tung Nan Institute of Technology, Taiwan, Republic of China

    Wen-Der Ueng

  2. Mechanical Engineering Department, National Central University, Taiwan, Republic of China

    Jiing-Yih Lai & Yao-Chen Tsai

Authors
  1. Wen-Der Ueng
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  2. Jiing-Yih Lai
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  3. Yao-Chen Tsai
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Correspondence to Jiing-Yih Lai.

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Ueng, WD., Lai, JY. & Tsai, YC. Unconstrained and constrained curve fitting for reverse engineering. Int J Adv Manuf Technol 33, 1189–1203 (2007). https://doi.org/10.1007/s00170-006-0557-8

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  • DOI: https://doi.org/10.1007/s00170-006-0557-8

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