Triangular Bernstein-Bézier patches

4. Hermite Interpolants . . , . . . . . . . . . . . . . 104 4.1. The Co nine parameter interpolant ...... 104 4.2. C’ quintic interpolants 104 4.3. The general case 106 5. Split Triangle Interpolants . . . . . . . . . 107 5.1. The C’ Clough-Tocher interpolant . . _ . . 108 5.2. Limitations of the Clough-Tocher split . 110 5.3. The C’ Powell-Sabin interpolants . . . . 111 5.4. C’ Split square interpolants . . . . . . . . . 112

[1]  Gerald Farin.  Piecewise triangular C1 surface strips , 1986 .

[2]  Peter Alfeld,et al.  A trivariate clough-tocher scheme for tetrahedral data , 1984, Comput. Aided Geom. Des..

[3]  Gary J. Herron,et al.  Smooth closed surfaces with discrete triangular interpolants , 1985, Comput. Aided Geom. Des..

[4]  I. Faux,et al.  Computational Geometry for Design and Manufacture , 1979 .

[5]  Tracy Whelan,et al.  A representation of a C2 interpolant over triangles , 1986, Comput. Aided Geom. Des..

[6]  Rita Schmidt,et al.  Eine Methode zur Konstruktion von C1-Flächen zur Interpolation Unregelmässig Verteilter Daten , 1982 .

[7]  P. Percell.  On Cubic and Quartic Clough–Tocher Finite Elements , 1976 .

[8]  N. S. Barnett,et al.  Private communication , 1969 .

[9]  Gerald Farin,et al.  SMOOTH INTERPOLATION TO SCATTERED 3D DATA. , 1983 .

[10]  L. L. Schumaker,et al.  Efficient evaluation of multivariate polynomials , 1986, Comput. Aided Geom. Des..

[11]  Ingrid Brueckner.  Construction of Bézier points of quadrilaterals from those of triangles , 1980 .

[12]  W. J. Gordon,et al.  Smooth interpolation in triangles , 1973 .

[13]  Tim N. T. Goodman.  Further variation diminishing properties of Bernstein polynomials on triangles , 1987 .

[14]  R. Goldman,et al.  Conversion from Be´zier rectangles to Be´zier triangles , 1987 .

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

[16]  T. Sederberg,et al.  Comparison of three curve intersection algorithms , 1986 .

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

[18]  Ingolf Grieger.  Geometry cells and surface definition by finite elements , 1985, Comput. Aided Geom. Des..

[19]  Robert E. Barnhill,et al.  Surfaces in computer aided geometric design: a survey with new results , 1985, Comput. Aided Geom. Des..

[20]  Gerald E. Farin,et al.  A construction for visualC1 continuity of polynomial surface patches , 1982, Comput. Graph. Image Process..

[21]  Gerald E. Farin,et al.  A modified Clough-Tocher interpolant , 1985, Comput. Aided Geom. Des..

[22]  Ming-Jun Lai,et al.  On Bivariate Vertex Splines , 1985 .

[23]  Peter Alfeld,et al.  A bivariate C2 Clough-Tocher scheme , 1984, Comput. Aided Geom. Des..

[24]  B. Barsky.  The beta-spline: a local representation based on shape parameters and fundamental geometric measures , 1981 .

[25]  R. Riesenfeld,et al.  A geometric proof for the variation diminishing property of B-spline approximation , 1983 .

[26]  Paul Sablonnière,et al.  Composite finite elements of class Ck , 1985 .

[27]  Robin Sibson,et al.  A Seamed Quadratic Element for Contouring , 1981, Comput. J..

[28]  Larry L. Schumaker,et al.  The dimension of bivariate spline spaces of smoothnessr for degreed≥4r+1 , 1987 .

[29]  Wolfgang Böhm,et al.  A survey of curve and surface methods in CAGD , 1984, Comput. Aided Geom. Des..

[30]  Tim N. T. Goodman,et al.  Variation diminishing properties of Bernstein polynomials on triangles , 1987 .

[31]  Robert E. Barnhill,et al.  Representation and Approximation of Surfaces , 1977 .

[32]  Gerald E. Farin,et al.  A construction for visual C1 continuity of polynomial surface patches , 1982, Comput. Graph. Image Process..

[33]  G. Farin,et al.  C1 quintic interpolation over triangles: Two explicit representations , 1981 .

[34]  P. M. Prenter.  Splines and variational methods , 1975 .

[35]  G. Farin,et al.  Bézier polynomials over triangles and the construction of piecewise Cr polynomials , 1980 .

[36]  M. Sabin.  The use of piecewise forms for the numerical representation of shape , 1976 .

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

[38]  Ron Goldman,et al.  Subdivision algorithms for Bézier triangles , 1983 .

[39]  G. Farin,et al.  Ann-dimensional Clough-Tocher interpolant , 1987 .

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

[41]  Wolfgang Dahmen,et al.  Subdivision algorithms for the generation of box spline surfaces , 1984, Comput. Aided Geom. Des..

[42]  Daniel J. Filip.  Adaptive subdivision algorithms for a set of Bezier triangles , 1986 .

[43]  A. Ženíšek.  Interpolation polynomials on the triangle , 1970 .

[44]  Carl S. Petersen.  Adaptive contouring of three-dimensional surfaces , 1984, Comput. Aided Geom. Des..