Convexity and Bernstein-Bézier Polynomials

Abstract The purpose of this paper is to summarize recent results on the convexity of functional and parametric Bernstein-Bezier surfaces.

[1]  C. D. Boor,et al.  B-Form Basics. , 1986 .

[2]  Gengzhe Chang,et al.  Converse theorems of convexity for Bernstein polynomials over triangles , 1990 .

[3]  Paul Sablonnière,et al.  Error Bounds for Hermite Interpolation by Quadratic Splines on an α-Triangulation , 1987 .

[4]  Zoltan J. Cendes,et al.  C1 Quadratic Interpolation over Arbitrary Point Sets , 1987, IEEE Computer Graphics and Applications.

[5]  Charles A. Micchelli,et al.  Some Remarks on Nonnegative Polynomials on Polyhedra , 1989 .

[6]  Thomas A. Grandine,et al.  On convexity of piecewise polynomial functions on triangulations , 1989, Comput. Aided Geom. Des..

[7]  John A. Roulier,et al.  Bézier curves of positive curvature , 1988, Comput. Aided Geom. Des..

[8]  W. Dahmen,et al.  Scattered data interpolation by bivariate C1-piecewise quadratic functions , 1990 .

[9]  M. Beska,et al.  Convexity and variation diminishing property of multidimensional Bernstein polynomials , 1989, Analysis in Theory and Applications.

[10]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

[11]  Chang-Zheng Zhou.  On the convexity of parametric Bézier triangular surfaces , 1990, Comput. Aided Geom. Des..

[12]  Geng-zhe Chang,et al.  An improved condition for the convexity of Bernstein-Bézier surfaces over triangles , 1984, Comput. Aided Geom. Des..

[13]  Eitan Lapidot.  Convexity preserving and predicting by Bernstein polynomials , 1984 .

[14]  Wolfgang Dahmen,et al.  Convexity and Bernstein polynomials on k -simploids , 1990 .

[15]  A. Sharma,et al.  Variation diminishing properties and convexity for the tensor product Bernstein operator , 1992 .

[16]  Tomas Sauer.  Multivariate Bernstein polynomials and convexity , 1991, Comput. Aided Geom. Des..

[17]  Eugene L. Allgower,et al.  Triangulations by Reflections with Applications to Approximation , 1978 .

[18]  Philip J. Davis,et al.  The convexity of Bernstein polynomials over triangles , 1984 .

[19]  Eitan Lapidot.  On generalized mid-point convexity , 1981 .

[20]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[21]  Josef Hoschek,et al.  Convexity and Variation Diminishing Property of Bernstein Polynomials over Triangles , 1985 .

[22]  T. Goodman.  Shape preserving representations , 1989 .

[23]  Zheng-bin Wang,et al.  An improved condition for the convexity and positivity of Bernstein-Bézier surfaces over triangles , 1988, Comput. Aided Geom. Des..