Comparison of interval methods for plotting algebraic curves

This paper compares the performance and efficiency of different function range interval methods for plotting f(x, y)=0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Homer form and centred form), an affine arithmetic method, a Bernstein coefficient method, Taubin's method, Rivlin's method, Gopalsamy's method, and related methods which also take into account derivative information. Our experimental results show that the affine arithmetic method, interval arithmetic using the centred form, the Bernstein coefficient method, Taubin's method, Rivlin's method, and their related derivative methods have similar performance, and generally they are more accurate and efficient than Gopalsamy's method and interval arithmetic using the power basis, the Bernstein basis, and Horner form methods.

[1]  Gabriel Taubin,et al.  Distance approximations for rasterizing implicit curves , 1994, TOGS.

[2]  Gabriel Taubin,et al.  Rasterizing algebraic curves and surfaces , 1994, IEEE Computer Graphics and Applications.

[3]  Ralph R. Martin,et al.  Interval and Affine Arithmetic for Surface Location of Power- and Bernstein-Form Polynomials , 2000, IMA Conference on the Mathematics of Surfaces.

[4]  Nicholas M. Patrikalakis,et al.  Robust interval solid modelling Part II: boundary evaluation , 1996, Comput. Aided Des..

[5]  Kevin G. Suffern,et al.  Interval methods in computer graphics , 1991, Comput. Graph..

[6]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[7]  Oved Shisha,et al.  The Bernstein form of a polynomial , 1966 .

[8]  Irina Voiculescu,et al.  Interval methods in geometric modeling , 2000, Proceedings Geometric Modeling and Processing 2000. Theory and Applications.

[9]  Jon G. Rokne,et al.  Computer Methods for the Range of Functions , 1984 .

[10]  Nicholas M. Patrikalakis,et al.  Approximation of measured data with interval B-splines , 1997, Comput. Aided Des..

[11]  Jorge Stolfi,et al.  Adaptive Enumeration of Implicit Surfaces with Affine Arithmetic , 1996, Comput. Graph. Forum.

[12]  Luiz Henrique de Figueiredo,et al.  Surface Intersection Using Affine Arithmetic , 1996, Graphics Interface.

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

[14]  Richard E. Chandler,et al.  A tracking algorithm for implicitly defined curves , 1988, IEEE Computer Graphics and Applications.

[15]  Jürgen Garloff.  Convergent Bounds for the Range of Multivariate Polynomials , 1985, Interval Mathematics.

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

[17]  Tom Duff,et al.  Interval arithmetic recursive subdivision for implicit functions and constructive solid geometry , 1992, SIGGRAPH.

[18]  J. Stolfi,et al.  Aane Arithmetic and Its Applications to Computer Graphics , 1990 .

[19]  T. J. Rivlin.  Bounds on a polynomial , 1970 .

[20]  Fujio Yamaguchi,et al.  Computer-Aided Geometric Design , 2002, Springer Japan.

[21]  Nicholas M. Patrikalakis,et al.  Robust interval solid modelling Part I: representations , 1996, Comput. Aided Des..

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

[23]  Philip S. Milne.  On the algorithms and implementation of a geometric algebra system , 1990 .

[24]  Adrian Bowyer,et al.  Robust arithmetic for multivariate Bernstein-form polynomials , 2000, Comput. Aided Des..

[25]  Fujio Yamaguchi,et al.  Curves and Surfaces in Computer Aided Geometric Design , 1988, Springer Berlin Heidelberg.

[26]  Rida T. Farouki,et al.  Approximation by interval Bezier curves , 1992, IEEE Computer Graphics and Applications.

[27]  John M. Snyder,et al.  Interval analysis for computer graphics , 1992, SIGGRAPH.

[28]  Irina Voiculescu,et al.  Affine arithmetic in matrix form for polynomial evaluation and algebraic curve drawing , 2002 .

[29]  Hans-Peter Seidel,et al.  Sampling procedural shaders using affine arithmetic , 1998, TOGS.

[30]  Kevin G. Suffern,et al.  Quadtree Algorithms for Contouring Functions of Two Variables , 1990, Comput. J..

[31]  Sudhir P. Mudur,et al.  A new method of evaluating compact geometric bounds for use in subdivision algorithms , 1991, Comput. Aided Geom. Des..

[32]  Hans-Peter Seidel,et al.  Sampling procedural shaders using affine arithmetic , 1997, SIGGRAPH '97.