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Convexity of parametric Bézier surfaces in terms of Gaussian curvature signatures

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Abstract

The shape of a smooth surfaceS in ℝ3 is governed by its Gaussian curvatureK. In this paper, the notion ofGC signature of a smooth surfaceS is introduced. This scalar-valued functionf has the same sign asK and becomes a polynomial ifS is a parametric polynomial surface. Hence, the study of convexity of such surfacesS (or nonparametric polynomialB) reduces to the study of positivity of the corresponding polynomialsf. Efficient computational schemes are developed for expressing the Bézier coefficients of the Bernstein-Bézier formulation off in terms of those of the parametric polynomialb (orB) that definesS. This approach differs from the usual consideration of positive definiteness of the Hessian matrix corresponding to the parametric polynomialb (orB).

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References

  1. R.E. Barnhill and T. Whelan, A geometric interpretation of convexity conditions for surfaces, Comp. Aided Geom. Design 1(1984)285–287.

    Article  MATH  Google Scholar 

  2. M.P. do Carmo,Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, NJ, 1976).

    MATH  Google Scholar 

  3. G.Z. Chang and P.J. Davis, The convexity of Bernstein polynomials over triangles, J. Approx. Theory 40(1984)11–28.

    Article  MATH  MathSciNet  Google Scholar 

  4. G.Z. Chang and Y.Y. Feng, An improved condition for the convexity of Bernstein-Bézier surfaces over triangles, Comp. Aided Geom. Design 1(1984)279–283.

    Article  MATH  Google Scholar 

  5. W. Dahmen, Convexity and Bernstein-Bézier polynomials, in:Curves and Surfaces, ed. P.L. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, Boston, 1991) pp. 107–134.

    Google Scholar 

  6. T.N.T. Goodman, Convexity of Bézier nets on triangulations, Comp. Aided Geom. Design 8(1991)175–180.

    Article  MATH  Google Scholar 

  7. C.A. Micchelli and A. Pinkus, Some remarks on nonnegative polynomials on polyhedra, in:Probability, Statistics, and Mathematics: Papers in Honor of Samual Karlin, ed. T.W. Anderson, K.B. Athreya and D.L. Iglehart (Academic Press, Boston, 1989).

    Google Scholar 

  8. H. Prautzsch, On convex triangles, Math. Mod. Numer. Anal. (1992)23–36.

  9. L.L. Schumaker and W. Volk, Efficient evaluation of multivariate polynomial, Comp. Aided Geom. Design 3(1986)149–154.

    Article  MATH  Google Scholar 

  10. Z.B. Wang and Q.M. Lin, An improved condition for the convexity and positivity of Bernstein-Bézier surfaces over triangles, Comp. Aided Geom. Design 5(1989)269–275.

    Article  MATH  MathSciNet  Google Scholar 

  11. Z. Xuang and G. Chang, Remarks on convexity for parametric Bézier triangular patches, Int. Center for Theoretical Physics, Report No. 10(1985) pp. 1–11.

    Google Scholar 

  12. E.C. Young,Vector and Tensor Analysis (Marcel Dekker, New York, 1978).

    MATH  Google Scholar 

  13. J.M. Zhao, Convexity theorem of parametric triangular Bézier surfaces, Chin. Ann. Math. 9(1988)134–145.

    MATH  Google Scholar 

  14. C.Z. Zhou, On the convexity of parametric Bézier triangular surfaces, Comp. Aided Geom. Deisgn 7(1990)459–436.

    Article  MATH  Google Scholar 

Download references

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Author notes

  1. Zhengxing Cheng

    Present address: Department of Mathematics, Xian Jiaotong University, Xian, P.R. China

Authors and Affiliations

  1. Center for Approximation Theory, Texas A&M University, 77843, College Station, TX, USA

    Zhengxing Cheng & Charles K. Chui

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  1. Zhengxing Cheng
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  2. Charles K. Chui
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Additional information

Research supported by NSF Grant DMS-92-06928 and ARO Contract DAAH 04-93-G-0047.

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Cheng, Z., Chui, C.K. Convexity of parametric Bézier surfaces in terms of Gaussian curvature signatures. Adv Comput Math 2, 437–459 (1994). https://doi.org/10.1007/BF02521608

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  • DOI: https://doi.org/10.1007/BF02521608

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