The Experts below are selected from a list of 5844 Experts worldwide ranked by 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, 2015Co-Authors: Rida T Farouki, Carlotta Giannelli, Alessandra SestiniAbstract: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, 2007Co-Authors: Christian Perwass, Rida T Farouki, Lyle NoakesAbstract: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, 2002Co-Authors: Rida T Farouki, Mohammad Alkandari, Takis SakkalisAbstract: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, 2002Co-Authors: Rida T Farouki, Mohammad Alkandari, Takis SakkalisAbstract: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, 2001Co-Authors: Hwan Pyo Moon, Rida T Farouki, Hyeong In ChoiAbstract: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.
A. Yu. Kamenshchik - One of the best experts on this subject based on the ideXlab platform.
-
Relativistic Hodograph equation for a two-dimensional stationary isentropic hydrodynamical motion
Physics Letters A, 2004Co-Authors: Isaak M. Khalatnikov, A. Yu. KamenshchikAbstract:We derive a relativistic Hodograph equation for a two-dimensional stationary isentropic hydrodynamical motion. For the case of stiff matter, when the velocity of sound coincides with the light speed, the singularity in this equation disappears and the solutions become regular in all Hodograph plane.
-
Relativistic Hodograph equation for a two-dimensional stationary isentropic hydrodynamical motion
Physics Letters A, 2004Co-Authors: Isaak M. Khalatnikov, A. Yu. KamenshchikAbstract:We derive a relativistic Hodograph equation for a two-dimensional stationary isentropic hydrodynamical motion. For the case of stiff matter, when the velocity of sound coincides with the light speed, the singularity in this equation disappears and the solutions become regular in all Hodograph plane.Comment: 5 page
Takis Sakkalis - One of the best experts on this subject based on the ideXlab platform.
-
hermite interpolation by rotation invariant spatial pythagorean Hodograph curves
Advances in Computational Mathematics, 2002Co-Authors: Rida T Farouki, Mohammad Alkandari, Takis SakkalisAbstract: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, 2002Co-Authors: Rida T Farouki, Mohammad Alkandari, Takis SakkalisAbstract: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.
-
pythagorean Hodograph space curves
Advances in Computational Mathematics, 1994Co-Authors: Rida T Farouki, Takis SakkalisAbstract:We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose Hodographs (derivatives) satisfy the Pythagorean conditionx′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-Hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for allcubic PythagoreanHodograph space curves, in terms of constraints on the Bezier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-Hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-Hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.
-
Pythagorean-Hodograph space curves
Advances in Computational Mathematics, 1994Co-Authors: Rida T Farouki, Takis SakkalisAbstract:We investigate the properties of polynomial space curves r(t) ={ x(t), y(t), z(t) } whose Hodographs (derivatives) satisfy the Pythagorean condition x ′^2( t )+ y ′^2( t )+ z ′^2( t )≡σ^2( t ) for some real polynomial σ( t ). The algebraic structure of the complete set of regular Pythagorean-Hodograph curves in ℝ^3 is inherently more complicated than that of the corresponding set in ℝ^2. We derive a characterization for all cubic PythagoreanHodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-Hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ^3), but they have no “simple” all-encompassing characterization. We focus on a subset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-Hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) the canal surfaces based on such curves as spines have precise rational parameterizations.
A V Desherevskii - One of the best experts on this subject based on the ideXlab platform.
-
testing rayleigh schuster Hodographs using time series models and earthquake flows
Seismic Instruments, 2016Co-Authors: A V Desherevskii, Ya A SidorinAbstract:The paper considers features and peculiarities of Rayleigh−Schuster Hodographs designed for detailed investigation of changes in the phase of quasiperiodic signals in time series. The method allows a researcher to estimate the statistical significance of analyzed periodicities and visualize a change in the signal phase. The method is tested using various models, in particular, flows of random events, earthquake flows, periodic signals, and noise-contaminated periodic signals of earthquake flows. The equations for estimating errors in the resulting vector phase are given. It is clearly demonstrated that the Hodographs of random signals can in some cases be similar to nonrandom ones. This effect contradicts intuitive concepts and creates the illusion of nonrandom signals. Possible causes of the phenomenon are discussed. A methodology for Hodograph analysis ensuring a more feasible interpretation is proposed.
-
Testing Rayleigh–Schuster Hodographs using time series models and earthquake flows
Seismic Instruments, 2016Co-Authors: A V Desherevskii, A. Ya. SidorinAbstract:The paper considers features and peculiarities of Rayleigh−Schuster Hodographs designed for detailed investigation of changes in the phase of quasiperiodic signals in time series. The method allows a researcher to estimate the statistical significance of analyzed periodicities and visualize a change in the signal phase. The method is tested using various models, in particular, flows of random events, earthquake flows, periodic signals, and noise-contaminated periodic signals of earthquake flows. The equations for estimating errors in the resulting vector phase are given. It is clearly demonstrated that the Hodographs of random signals can in some cases be similar to nonrandom ones. This effect contradicts intuitive concepts and creates the illusion of nonrandom signals. Possible causes of the phenomenon are discussed. A methodology for Hodograph analysis ensuring a more feasible interpretation is proposed.
-
improvement of robustness and stability in estimating rayleigh schuster s Hodograph parameters using different procedures of vector normalization
Seismic Instruments, 2016Co-Authors: A V Desherevskii, Ya A SidorinAbstract:This paper deals with Rayleigh–Schuster’s Hodographs, intended for the detailed investigation of changes in the phase of quasiperiodic signals in time series. The Hodographs are also known as the phasor-walkout method. A procedure of conditional vector normalization is proposed: it takes into account the vector amplitude for each period under consideration. The procedure considerably improves the robustness and stability of the Hodograph approach to changes in the character of the processed data distributions and to various defects in the data. For example, when analyzing the earthquake catalog, the procedure strongly diminishes the influence of the event clustering caused, in particular, by swarms of earthquakes with comparable magnitude and the aftershock sequences of strong earthquakes. At the first stage, we calculate the vector sums (resulting vectors) for each period under investigation throughout the time series duration. For example, investigating diurnal periodicity of earthquakes, we first calculate the resulting vectors for each day of the observation. The further analysis of resulting vectors for each period throughout the time series duration can be performed with different procedures. We compare three procedures for normalization of the obtained resulting vectors, which are as follows: (1) th traditional one, preserving the real signal amplitude; (2) that with reduction of the obtained resulting vectors to the unit vector (phasor); and (3) that with conditional vector normalization, taking into account the amplitude of resulting vectors for each period throughout the time series duration. The third procedure diminishes the possible instability in some special distributions of the investigated data when the resulting vector for a period is close to zero. The procedures are compared using model signals and samples from real earthquake catalogs. All the procedures used give close results when processing random time series.
Ya A Sidorin - One of the best experts on this subject based on the ideXlab platform.
-
testing rayleigh schuster Hodographs using time series models and earthquake flows
Seismic Instruments, 2016Co-Authors: A V Desherevskii, Ya A SidorinAbstract:The paper considers features and peculiarities of Rayleigh−Schuster Hodographs designed for detailed investigation of changes in the phase of quasiperiodic signals in time series. The method allows a researcher to estimate the statistical significance of analyzed periodicities and visualize a change in the signal phase. The method is tested using various models, in particular, flows of random events, earthquake flows, periodic signals, and noise-contaminated periodic signals of earthquake flows. The equations for estimating errors in the resulting vector phase are given. It is clearly demonstrated that the Hodographs of random signals can in some cases be similar to nonrandom ones. This effect contradicts intuitive concepts and creates the illusion of nonrandom signals. Possible causes of the phenomenon are discussed. A methodology for Hodograph analysis ensuring a more feasible interpretation is proposed.
-
improvement of robustness and stability in estimating rayleigh schuster s Hodograph parameters using different procedures of vector normalization
Seismic Instruments, 2016Co-Authors: A V Desherevskii, Ya A SidorinAbstract:This paper deals with Rayleigh–Schuster’s Hodographs, intended for the detailed investigation of changes in the phase of quasiperiodic signals in time series. The Hodographs are also known as the phasor-walkout method. A procedure of conditional vector normalization is proposed: it takes into account the vector amplitude for each period under consideration. The procedure considerably improves the robustness and stability of the Hodograph approach to changes in the character of the processed data distributions and to various defects in the data. For example, when analyzing the earthquake catalog, the procedure strongly diminishes the influence of the event clustering caused, in particular, by swarms of earthquakes with comparable magnitude and the aftershock sequences of strong earthquakes. At the first stage, we calculate the vector sums (resulting vectors) for each period under investigation throughout the time series duration. For example, investigating diurnal periodicity of earthquakes, we first calculate the resulting vectors for each day of the observation. The further analysis of resulting vectors for each period throughout the time series duration can be performed with different procedures. We compare three procedures for normalization of the obtained resulting vectors, which are as follows: (1) th traditional one, preserving the real signal amplitude; (2) that with reduction of the obtained resulting vectors to the unit vector (phasor); and (3) that with conditional vector normalization, taking into account the amplitude of resulting vectors for each period throughout the time series duration. The third procedure diminishes the possible instability in some special distributions of the investigated data when the resulting vector for a period is close to zero. The procedures are compared using model signals and samples from real earthquake catalogs. All the procedures used give close results when processing random time series.