Bézier Techniques

This chapter provides a thorough review of fundamental Bezier Techniques as a core tool of 3D modeling or computer aided geometric design (CAGD). Bezier techniques provide a geometric-based method for describing and manipulating polynomial curves and surfaces. Bezier techniques bring sophisticated mathematical concepts into a highly geometric and intuitive form. This form facilitates the creative design process. Bezier techniques are used in the context of numerical stability of floating point operations. The chapter describes curves, rectangular surfaces, and triangular surfaces. It introduces the foundations of Bezier techniques via the Bezier curve. It presents the properties and utility of Bezier curves, as well as an evaluation algorithm: the de Casteljau algorithm. The de Casteljau algorithm provides a means for evaluating Bezier curves. It also provides for greater understanding of Bezier methods as a whole. The chapter also discusses the building block of Bezier techniques: the Bernstein polynomials. It describes various properties of Bezier patches, such as endpoint interpolation, symmetry, affine invariance, bilinear precision, tensor product, and functional patches.

[1]  Gerald E. Farin,et al.  A geometric interpretation of the diagonal of a tensor-product Bézier volume , 1999, Comput. Aided Geom. Des..

[2]  Wolfgang Dahmen,et al.  Subdivision algorithms converge quadratically , 1986 .

[3]  Gian-Carlo Rota,et al.  Linear Operators and Approximation Theory. , 1965 .

[4]  John A. Gregory,et al.  Convexity of Bézier nets on sub-triangles , 1991, Comput. Aided Geom. Des..

[5]  R. Goldman.  Using degenerate Bézier triangles and tetrahedra to subdivide Bézier curves , 1982 .

[6]  A. R. Forrest,et al.  The twisted cubic curve: a computer-aided geometric design approach , 1980 .

[7]  Ronald Goldman,et al.  An urnful of blending functions , 1983, IEEE Computer Graphics and Applications.

[8]  Günter Aumann,et al.  Interpolation with developable Bézier patches , 1991, Comput. Aided Geom. Des..

[9]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[10]  A. Robin Forrest,et al.  Interactive interpolation and approximation by Bezier polynomials , 1972, Comput. J..

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

[12]  J. Hoschek.  Offset curves in the plane , 1985 .

[13]  Gerald Farin,et al.  Link between Be´zier and lagrange curve and surface schemes , 1986 .

[14]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[15]  Klaus Spitzmüller,et al.  Technical note Partial derivatives of Bbzier surfaces , 2003 .

[16]  P. Barry,et al.  Three examples of dual properties of Be´zier curves , 1989 .

[17]  Gerald E. Farin,et al.  The essentials of CAGD , 2000 .

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

[19]  Paul de Faget de Casteljau.  De Casteljau's autobiography: My time at Citroën , 1999, Comput. Aided Geom. Des..

[20]  Hartmut Prautzsch,et al.  Arbitrarily high degree elevation of Bézier representations , 1996, Comput. Aided Geom. Des..

[21]  Juan Manuel Peña,et al.  Linear convexity conditions for rectangular and triangular Bernstein-Bézier surfaces , 1997, Comput. Aided Geom. Des..

[22]  Natasha Dejdumrong,et al.  Efficient algorithms for Bézier curves , 2000, Comput. Aided Geom. Des..

[23]  Elaine COHEN,et al.  Rates of convergence of control polygons , 1985, Comput. Aided Geom. Des..

[24]  Richard F. Riesenfeld,et al.  A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[25]  Yap Siong Chua.  Bézier brushstrokes , 1990, Comput. Aided Des..

[26]  Nickolas S. Sapidis,et al.  Visualization of curvature plots and evaluation of fairness: an analysis of the effect of 'scaling' , 1997, Comput. Aided Geom. Des..

[27]  P. Bézier.  Numerical control : mathematics and applications , 1972 .

[28]  Dieter Lasser,et al.  Grundlagen der geometrischen Datenverarbeitung , 1989 .

[29]  A. W. Nutbourne,et al.  Curvature profiles for plane curves , 1972, Comput. Aided Des..

[30]  Xuguang Wang,et al.  Rational hodographs , 1987, Comput. Aided Geom. Des..

[31]  Alan Pipes.  Interactive techniques in CAD: 21–23 September 1978, Palazzo dei Congressi, Bologna, Italy. Organized by ACM Italian Chapter , 1979 .

[32]  Matthias Eck,et al.  Least squares degree reduction of Bézier curves , 1995, Comput. Aided Des..

[33]  Marc Daniel,et al.  The numerical problem of using Bézier curves and surfaces in the power basis , 1989, Comput. Aided Geom. Des..

[34]  Ron Goldman,et al.  Linear subdivision is strictly a polynomial phenomenon , 1984, Comput. Aided Geom. Des..

[35]  Wolfgang Böhm,et al.  On cubics: A survey , 1982, Comput. Graph. Image Process..

[36]  Yongming Li,et al.  Basis conversion among Bézier, Tchebyshev and Legendre , 1998, Comput. Aided Geom. Des..

[37]  Yu-Yu Feng,et al.  Rates of convergence of Bézier net over triangles , 1987, Comput. Aided Geom. Des..

[38]  P. Bézier.  General distortion of an ensemble of biparametric surfaces , 1978 .

[39]  Rida T. Farouki,et al.  A Hierarchy of Geometric Forms , 1985, IEEE Computer Graphics and Applications.

[40]  Wendelin L. F. Degen.  The Types of Triangular Bézier Surfaces , 1994, IMA Conference on the Mathematics of Surfaces.

[41]  Gerald Farin,et al.  SOME ASPECTS OF CAR BODY DESIGN AT DAIMLER-BENZ. , 1983 .

[42]  Wendelin L. F. Degen,et al.  Some remarks on Bézier curves , 1988, Comput. Aided Geom. Des..

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

[44]  Thomas W. Sederberg,et al.  Free-form deformation of solid geometric models , 1986, SIGGRAPH.

[45]  Rida T. Farouki,et al.  On the numerical condition of polynomials in Bernstein form , 1987, Comput. Aided Geom. Des..

[46]  Wolfgang Böhm,et al.  On de Casteljau's algorithm , 1999, Comput. Aided Geom. Des..

[47]  Wolfgang Böhm.  Sudividing multivariate splines , 1983 .

[48]  Tony DeRose,et al.  A Tutorial Introduction to Blossoming , 1991 .

[49]  Dieter Lasser,et al.  Intersection of parametric surfaces in the Bernstein—Be´zier representation , 1986 .

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

[51]  Faget de Casteljau,et al.  Shape mathematics and CAD , 1986 .

[52]  W. Böhm,et al.  Generating the Bézier points of B-spline curves and surfaces , 1981 .

[53]  P. Barry,et al.  What is the natural generalization of a Be´zier curve? , 1989 .

[54]  Jörg Peters,et al.  Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon , 1999, Comput. Aided Geom. Des..

[55]  Wolfgang Böhm.  Bézier presentation of airfoils , 1987, Comput. Aided Geom. Des..

[56]  Martin Ahlers,et al.  Role of Bézier curves and surfaces in the volkswagen CAD approach from 1967 to today , 1990, Comput. Aided Des..

[57]  Thomas A. Foley,et al.  The hybrid quintic Bézier tetrahedron , 1997, Comput. Aided Geom. Des..

[58]  Guido Brunnett,et al.  The geometry of optimal degree reduction of Bézier curves , 1996, Comput. Aided Geom. Des..

[59]  Dieter Lasser,et al.  Bernstein-Bézier representation of volumes , 1985, Comput. Aided Geom. Des..

[60]  Helmut Pottmann,et al.  Developable rational Bézier and B-spline surfaces , 1995, Comput. Aided Geom. Des..

[61]  Louis J. Nachman.  A note on control polygons and derivatives , 1991, Comput. Aided Geom. Des..

[62]  Thomas Reuding,et al.  Bézier patches on cubic grid curves - An application to the preliminary design of a yacht hull surface , 1989, Comput. Aided Geom. Des..

[63]  Nadim M. Aziz,et al.  Bezier surface/surface intersection , 1990, IEEE Computer Graphics and Applications.

[64]  A. J. Worsey,et al.  Degree reduction of Be´zier curves , 1988 .

[65]  Hans Hagen.  Bezier-curves with curvature and torsion continuity , 1986 .

[66]  Matthias Eck,et al.  Degree reduction of Bézier curves , 1993, Comput. Aided Geom. Des..

[67]  Tony DeRose,et al.  Computing values and derivatives of Bézier and B-spline tensor products , 1995, Comput. Aided Geom. Des..

[68]  H. P. Seidel.  A general subdivision theorem for Be´zier triangles , 1989 .

[69]  P. Chambers,et al.  Interactive curves and surfaces , 1996, Technology-Based Re-Engineering Engineering Education Proceedings of Frontiers in Education FIE'96 26th Annual Conference.

[70]  Carlo H. Séquin,et al.  Local surface interpolation with Bézier patches: errata and improvements , 1991, Comput. Aided Geom. Des..

[71]  D. Hilbert,et al.  Geometry and the Imagination , 1953 .

[72]  Rida T. Farouki,et al.  Algorithms for polynomials in Bernstein form , 1988, Comput. Aided Geom. Des..

[73]  J. Gallier.  Curves and surfaces in geometric modeling: theory and algorithms , 1999 .

[74]  Eberhard Eisele,et al.  Best approximations of symmetric surfaces by biquadratic Bézier surfaces , 1994, Comput. Aided Geom. Des..

[75]  Jörg Peters,et al.  Polynomial degree reduction in the L2-norm equals best Euclidean approximation of Bézier coefficients , 1999, Comput. Aided Geom. Des..

[76]  B. Barsky,et al.  An Introduction to Splines for Use in Computer Graphics and Geometric Modeling , 1987 .

[77]  Ljubiša Kocić.  Modification of Bézier curves and surfaces by degree-elevation technique , 1991 .

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

[79]  Ron Goldman,et al.  Recursive subdivision without the convex hull property , 1986, Comput. Aided Geom. Des..

[80]  D. Marsh.  Applied Geometry for Computer Graphics and CAD , 1999 .

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

[82]  Gerald E. Farin,et al.  Curvature and the fairness of curves and surfaces , 1989, IEEE Computer Graphics and Applications.

[83]  P. E. Bezier,et al.  Example of an existing system in the motor industry: the Unisurf system , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[84]  A. Robin Forrest,et al.  Interactive interpolation and approximation by Bézier polynomials , 1990, Comput. Aided Des..

[85]  Rida T. Farouki,et al.  On the stability of transformations between power and Bernstein polynomial forms , 1991, Comput. Aided Geom. Des..

[86]  Lyle Ramshaw,et al.  Béziers and B-splines as Multiaffine Maps , 1988 .

[87]  Shi-Min Hu.  Conversion of a tringular Bézier patch into three rectangular Bézier patches , 1996, Comput. Aided Geom. Des..