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Marie Laurence Mazure - One of the best experts on this subject based on the ideXlab platform.

  • the fundamental blossoming inequality in chebyshev spaces i applications to schur functions
    Foundations of Computational Mathematics, 2018
    Co-Authors: Rachid Aithaddou, Marie Laurence Mazure
    Abstract:

    A classical theorem by Chebyshev says how to obtain the minimum and maximum values of a symmetric multiaffine function of n variables with a prescribed sum. We show that, given two functions in an Extended Chebyshev space good for design, a similar result can be stated for the minimum and maximum values of the blossom of the first function with a prescribed value for the blossom of the second one. We give a simple geometric condition on the Control Polygon of the planar parametric curve defined by the pair of functions ensuring the uniqueness of the solution to the corresponding optimization problem. This provides us with a fundamental blossoming inequality associated with each Extended Chebyshev space good for design. This inequality proves to be a very powerful tool to derive many classical or new interesting inequalities. For instance, applied to Muntz spaces and to rational Muntz spaces, it provides us with new inequalities involving Schur functions which generalize the classical MacLaurin’s and Newton’s inequalities. This work definitely demonstrates that, via blossoms, CAGD techniques can have important implications in other mathematical domains, e.g., combinatorics.

Xuli Han - One of the best experts on this subject based on the ideXlab platform.

Ahmad H Nasri - One of the best experts on this subject based on the ideXlab platform.

  • a subdivision algorithm for generating rational curves
    Journal of Graphics Tools, 2002
    Co-Authors: Ahmad H Nasri, Gerald Farin
    Abstract:

    The well-known Chaikin algorithm generates uniform quadratic B-spline curves by repeating the process of cutting off the corners of a Polygon. One disadvantage of this algorithm is the incapability of generating circles. This paper proposes a modification of this algorithm to produce piecewise rational curves; in particular a circle is produced from a given square. For a general Control Polygon, every two subsequent Polygon legs of equal length will correspond to a circular arc. Such an arc will be parameterized by arc length and will remain circular under affine transformations. Both properties are not shared by the standard rational quadratic form.

  • a recursive subdivision algorithm for piecewise circular spline
    Computer Graphics Forum, 2001
    Co-Authors: Ahmad H Nasri, C W A M Van Overveld, Brian Wyvill
    Abstract:

    We present an algorithm for generating a piecewise G1 circular spline curve from an arbitrary given Control Polygon. For every corner, a circular biarc is generated with each piece being parameterized by its arc length. This is the first subdivision scheme that produces a piecewise biarc curve that can interpolate an arbitrary set of points. It is easily adopted in a recursive subdivision surface scheme to generate surfaces with circular boundaries with pieces parameterized by arc length, a property not previously available. As an application, a modified version of Doo–Sabin subdivision algorithm is outlined making it possible to blend a subdivision surface with other surfaces having circular boundaries such as cylinders.

  • an algorithm for interpolating intersecting curves by recursive subdivision surfaces
    International Conference on Shape Modeling and Applications, 1999
    Co-Authors: Ahmad H Nasri
    Abstract:

    Interpolation conditions on recursive subdivision surfaces provide more powerful techniques to manipulate such surfaces. Recently, such conditions were extended to handle interpolation of pre-defined curves by a subdivision surface. Given a curve C/sub i/ defined by a Control Polygon cp/sub 0/, this consists of constructing a strip complex P/sub i/ as part of the defining polyhedral network M/sub 0/ or its first subdivision M/sub 1/. By repeated subdivision, M/sub i/ converges to a limit surface S which interpolates the curve C/sub i/. We describe an algorithm for constructing strip complexes that interpolate intersecting curves at the boundary of a surface. The algorithm is an important step towards solving the problem of interpolating arbitrary intersecting meshes of curves by subdivision surfaces.

  • curve interpolation in recursively generated b spline surfaces over arbitrary topology
    Computer Aided Geometric Design, 1997
    Co-Authors: Ahmad H Nasri
    Abstract:

    Abstract Recursive subdivision is receiving a great deal of attention in the definition of B-spline surfaces over arbitrary topology. The technique has recently been extended to generate interpolating surfaces with given normal vectors at the interpolated vertices. This paper describes an algorithm to generate recursive subdivision surfaces that interpolate B-spline curves. The Control Polygon of each curve is defined by a path of vertices of the polyhedral network describing the surface. The method consists of applying a one-step subdivision of the initial network and modifying the topology in the neighborhood of the vertices generated from the Control Polygons. Subsequent subdivisions of the modified network generate sequences of Polygons each of which converges to a curve interpolated by the limit surface. In the case of regular networks, the method can be reduced to a knot insertion process.

Ghulam Mustafa - One of the best experts on this subject based on the ideXlab platform.

  • Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form
    'Hindawi Limited', 2021
    Co-Authors: Rabia Hameed, Ghulam Mustafa, Jiansong Deng, Shafqat Ali
    Abstract:

    In this article, we present a new method to construct a family of 2N+2-point binary subdivision schemes with one tension parameter. The construction of the family of schemes is based on repeated local translation of points by certain displacement vectors. Therefore, refinement rules of the 2N+2-point schemes are recursively obtained from refinement rules of the 2N-point schemes. Thus, we get a new subdivision scheme at each iteration. Moreover, the complexity, polynomial reproduction, and polynomial generation of the schemes are increased by two at each iteration. Furthermore, a family of interproximate subdivision schemes with tension parameters is also introduced which is the extended form of the proposed family of schemes. This family of schemes allows a different tension value for each edge and vertex of the initial Control Polygon. These schemes generate curves and surfaces such that some initial Control points are interpolated and others are approximated

  • a new computational approach to estimate the subdivision depth of n ary subdivision scheme
    IEEE Access, 2020
    Co-Authors: Ghulam Mustafa, Faheem Khan, Dumitru Baleanu, Aamir Shahzad, Yuming Chu
    Abstract:

    The $n$ -ary subdivision scheme has traditionally been designed to generate smooth curve and surface from Control Polygon. In this paper, we propose a new subdivision depth computation technique for $n$ -ary subdivision scheme. The existing techniques do not ensure the computation of subdivision depth unless some strong condition is assumed on the mask of the scheme. But our technique relaxes the effect of strong condition assumed on the mask of the scheme by increasing the number of convolution steps. Consequently, a more precise subdivision depth technique for a given error tolerance is presented in this paper.

  • recursive process for constructing the refinement rules of new combined subdivision schemes and its extended form
    arXiv: Numerical Analysis, 2018
    Co-Authors: Rabia Hameed, Ghulam Mustafa
    Abstract:

    In this article, we present a new method to construct a family of (2N+2)-point binary subdivision schemes with one tension parameter where N is a non-negative integer. The construction of the family of schemes is based on repeated local translation of points by certain displacement vectors. Therefore, the refinement rules of a (2N+2)-point scheme for N=M are recursively obtained from the refinement rules of the (2N+2)-point schemes for N=0,1,2,...,M-1. The complexity, polynomial reproduction and polynomial generation of these schemes are increased by two for the successive values of $N$. Furthermore, we modify this family of schemes to a family of (2N+3)-point schemes with two tension parameters. Moreover, a family of interproximate subdivision schemes with tension parameters is also introduced, which allows a different tension value for each edge and vertex of the initial Control Polygon. Interproximate schemes generate curves and surfaces such that some initial Control points are interpolated and others are approximated.

Rachid Aithaddou - One of the best experts on this subject based on the ideXlab platform.

  • the fundamental blossoming inequality in chebyshev spaces i applications to schur functions
    Foundations of Computational Mathematics, 2018
    Co-Authors: Rachid Aithaddou, Marie Laurence Mazure
    Abstract:

    A classical theorem by Chebyshev says how to obtain the minimum and maximum values of a symmetric multiaffine function of n variables with a prescribed sum. We show that, given two functions in an Extended Chebyshev space good for design, a similar result can be stated for the minimum and maximum values of the blossom of the first function with a prescribed value for the blossom of the second one. We give a simple geometric condition on the Control Polygon of the planar parametric curve defined by the pair of functions ensuring the uniqueness of the solution to the corresponding optimization problem. This provides us with a fundamental blossoming inequality associated with each Extended Chebyshev space good for design. This inequality proves to be a very powerful tool to derive many classical or new interesting inequalities. For instance, applied to Muntz spaces and to rational Muntz spaces, it provides us with new inequalities involving Schur functions which generalize the classical MacLaurin’s and Newton’s inequalities. This work definitely demonstrates that, via blossoms, CAGD techniques can have important implications in other mathematical domains, e.g., combinatorics.