The Experts below are selected from a list of 273 Experts worldwide ranked by ideXlab platform

A. Yu. Kamenshchik - One of the best experts on this subject based on the ideXlab platform.

Rida T Farouki - One of the best experts on this subject based on the ideXlab platform.

  • Identification and reverse engineering of Pythagorean-hodograph curves
    Computer Aided Geometric Design, 2015
    Co-Authors: Rida T Farouki, Carlotta Giannelli, Alessandra Sestini
    Abstract:

    Methods are developed to identify whether or not a given polynomial curve, specified by Bezier control points, is a Pythagorean-hodograph (PH) curve - and, if so, to reconstruct the internal algebraic structure that allows one to exploit the advantageous properties of PH curves. Two approaches to identification of PH curves are proposed. The first is based on the satisfaction of a system of algebraic constraints by the control-polygon legs, and the second uses the fact that numerical quadrature rules that are exact for polynomials of a certain maximum degree generate arc length estimates for PH curves exhibiting a sharp saturation as the number of sample points is increased. These methods are equally applicable to planar and spatial PH curves, and are fully elaborated for cubic and quintic PH curves. The reverse engineering problem involves computing the complex or quaternion coefficients of the pre-image polynomials generating planar or spatial Pythagorean Hodographs, respectively, from prescribed Bezier control points. In the planar case, a simple closed-form solution is possible, but for spatial PH curves the reverse engineering problem is much more involved. Methods to identify whether or not given control points define a Pythagorean-hodograph (PH) curve are formulated.The methods are based on the satisfaction of control-point constraints or saturation of quadrature arc-length estimates, and apply equally to planar and spatial PH curves.For identified PH curves, algorithms to reconstruct the complex or quaternion pre-image polynomials are developed.The proposed methods allow existing CAD systems to fully exploit the advantageous properties of PH curves within the context of prevailing CAD geometry representations.

  • a geometric product formulation for spatial pythagorean hodograph curves with applications to hermite interpolation
    Computer Aided Geometric Design, 2007
    Co-Authors: Christian Perwass, Rida T Farouki, Lyle Noakes
    Abstract:

    A novel formulation for spatial Pythagorean hodograph (PH) curves, based on the geometric product of vectors from Clifford algebra, is proposed. Compared to the established quaternion representation, in which a hodograph is generated by a continuous sequence of scalings/rotations of a fixed unit vector [email protected]?, the new representation corresponds to a sequence of scalings/reflections of [email protected]?. The two representations are shown to be equivalent for cubic and quintic PH curves, when freedom in choosing [email protected]? is retained for the vector formulation. The latter also subsumes the original (sufficient) characterization of spatial Pythagorean Hodographs, proposed by Farouki and Sakkalis, as a particular choice for [email protected]?. In the context of the spatial PH quintic Hermite interpolation problem, variation of the unit vector [email protected]? offers a geometrically more-intuitive means to explore the two-parameter space of solutions than the two free angular variables that arise in the quaternion formulation. This space is seen to have a decomposition into a product of two one-parameter spaces, in which one parameter determines the arc length and the other can be used to vary the curve shape at fixed arc length.

  • hermite interpolation by rotation invariant spatial pythagorean hodograph curves
    Advances in Computational Mathematics, 2002
    Co-Authors: Rida T Farouki, Mohammad Alkandari, Takis Sakkalis
    Abstract:

    The interpolation of first-order Hermite data by spatial Pythagorean-hodograph curves that exhibit closure under arbitrary 3-dimensional rotations is addressed. The Hodographs of such curves correspond to certain combinations of four polynomials, given by Dietz et al. [4], that admit compact descriptions in terms of quaternions – an instance of the “PH representation map” proposed by Choi et al. [2]. The lowest-order PH curves that interpolate arbitrary first-order spatial Hermite data are quintics. It is shown that, with PH quintics, the quaternion representation yields a reduction of the Hermite interpolation problem to three “simple” quadratic equations in three quaternion unknowns. This system admits a closed-form solution, expressing all PH quintic interpolants to given spatial Hermite data as a two-parameter family. An integral shape measure is invoked to fix these two free parameters.

  • structural invariance of spatial pythagorean Hodographs
    Computer Aided Geometric Design, 2002
    Co-Authors: Rida T Farouki, Mohammad Alkandari, Takis Sakkalis
    Abstract:

    The structural invariance of the four-polynomial characterization for three-dimensional Pythagorean Hodographs introduced by Dietz et al. (1993), under arbitrary spatial rotations, is demonstrated. The proof relies on a factored-quaternion representation for Pythagorean Hodographs in three-dimensional Euclidean space--a particular instance of the "PH representation map" proposed by Choi et al. (2002)--and the unit quaternion description of spatial rotations. This approach furnishes a remarkably simple derivation for the polynomials u'(t), v'(t), p'(t), q'(t) that specify the canonical form of a rotated Pythagorean hodograph, in terms of the original polynomials u(t), v(t), p(t), q(t) and the angle θ and axis n of the spatial rotation. The preservation of the canonical form of PH space curves under arbitrary spatial rotations is essential to their incorporation into computer-aided design and manufacturing applications, such as the contour machining of free-form surfaces using a ball-end mill and real-time PH curve CNC interpolators.

  • construction and shape analysis of ph hermite interpolants
    Computer Aided Geometric Design, 2001
    Co-Authors: Hwan Pyo Moon, Rida T Farouki, Hyeong In Choi
    Abstract:

    Abstract In general, the problem of interpolating given first-order Hermite data (end points and derivatives) by quintic Pythagorean-hodograph (PH) curves has four distinct formal solutions. Ordinarily, only one of these interpolants is of acceptable shape. Previous interpolation algorithms have relied on explicitly constructing all four solutions, and invoking a suitable measure of shape—e.g., the absolute rotation index or elastic bending energy—to select the “good” interpolant. We introduce here a new means to differentiate among the solutions, namely, the winding number of the closed loop formed by a union of the Hodographs of the PH quintic and of the unique “ordinary” cubic interpolant. We also show that, for “reasonable” Hermite data, the good PH quintic can be directly constructed with certainty, obviating the need to compute and compare all four solutions. Finally, we present an algorithm based on the subdivision, degree elevation, and convex hull properties of the Bernstein form, that gives rapidly convergent curvature bounds for PH curves, using only rational arithmetic operations on their coefficients.

M. S. Dudarev - One of the best experts on this subject based on the ideXlab platform.

Isaak M. Khalatnikov - One of the best experts on this subject based on the ideXlab platform.

Stanislav O Yurchenko - One of the best experts on this subject based on the ideXlab platform.

  • colloids in rotating electric and magnetic fields designing tunable interactions with spatial field Hodographs
    Soft Matter, 2020
    Co-Authors: Kirill A Komarov, Stanislav O Yurchenko
    Abstract:

    Opening a way to designing tunable interactions between colloidal particles, rotating electric and magnetic fields provide rich opportunities both for fundamental studies of phase transitions and engineering of soft materials. Spatial Hodographs, showing distribution of the field magnitude and orientation, allow to adjust the interactions and can be extremely potent tool for prospective experiments, but remain unstudied systematically. Here, we calculated the tunable interactions between spherical particles in rhodonea, conical, cylindrical, and ellipsoidal field Hodographs, as the most experimentally-important cases. We discovered that the spatial Hodographs are reduced to each other, providing a plethora of interactions, e.g., repulsive, attractive, barrier-like, and double-scale repulsive ones. Complementing the ``magic'' conical angle, the ``magic'' compression and ellipticity of cylindrical and ellipsoidal Hodographs are introduced. In the ``magic'' Hodographs, the interactions become spatially-isotropic and attain dispersion-force-like asymptotic (the same for pairwise and many-body energies), being attractive or repulsive, if the particle permittivity is larger or smaller than that of the solvent. With the diagrammatic method and numerical calculations, we obtained physically-meaningful fits to the many-body tunable potentials for silica (iron oxide) particles in deionised water in the rotating electric (magnetic) fields. Our results provide essential guidance for future experiments and simulations of colloidal liquids, crystals, gels, and glasses, important for broad range of problems in condensed matter, chemical physics, physical chemistry, materials science, and soft matter.