Rational Parametric Equations Research Articles

COMPOSITE GEOMETRICALLY SMOOTH CUBIC HERMITE CURVES IN SPACE AND ON THE PLANE

The article considers constructive algorithms for creating compound curves using parametric cubic equations. The possibilities of modern computer technology and systems of computer mathematics make it possible to implement algorithms of graph analytics using fractional rational parametric equations of the fifth, seventh and higher orders. The article considers a constructive algorithm for creating a compound cubic curve passing through given points with direction vectors specified in these points. Direction vectors can be used for monitoring the shape of a constructed curve. At the junction points of the cubic curve segments, second order geometric flatness is ensured, due to the equicontinuity of the camber and curvature. The required curve is made sequentially by attaching a new segment to the previous segment. Each segment must comply with the same group of geometric conditions: frequency at boundary points, contact with direction vectors, fixed curvature at one or both boundary points. The article proposes a constructive algorithm for constructing a three-dimensional or flat cubic Hermite segment that satisfies this group of conditions. The problem of constructing a compound geometrically smooth curve boils down to sequential construction of cubic segments using the proposed algorithm. The method proposed in the article for constructing a smooth compound curve allows: to shape a spatial curve with a given curvature at the starting point; to construct a plane curve with given radii of curvature at the nodes (in particular, with zero curvature at the nodal points); insert a cubic segment into a break in a curved line; find a smooth (without a break in a curve) conjugation of curvilinear and rectilinear sections of the constructed line. A distinctive feature of the algorithm is the significant use of reference tools and design capabilities of three-dimensional computer graphics. All necessary actions are performed using a desktop calculator and any software product that supports three-dimensional graphics (Compass-3D, AutoCAD, etc.). Five transparent examples of constructing flat and spatial composite cubic curves of the second order of smoothness are considered.

Forecasting of aerodynamic coefficients of tri-axially symmetrical Y plan shaped tall building based on CFD data trained ANN

From Burj Khalifa to Jeddah Tower, the use of a triaxially symmetrical plan is very common for skyscrapers. Corner modification is inevitable to decrease the wind-induced load, motion, and vibration, at least on the upper floors. This study focuses on forecasting force, moment and torsional coefficients of triaxially symmetrical Y plan shaped tall building. Initially, the Computational Fluid Dynamics (CFD) study of building models is done on the RANS k −ε turbulence model. The variation of aerodynamic coefficients with the corner cut percentage and angle of attack (AOA) is discussed. The flow patterns are also utilised for analysing the variation in the coefficient values. The design parameters and results are used for predicting some rational parametric equations of the aerodynamic coefficients. Outcomes from CFD are further utilised for the training of Artificial Neural Networks (ANN). Comparison of the results from CFD, ANN and rational parametric equations are made, and wind tunnel results are used for the validation purposes of the surrogate modelling. The maximum observed error for ANN modelling is less than 4% which suggests very good predictability of the networks.

Unirational differential curves and differential rational parametrizations

In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore, for linear differential curves, we give an algorithm to determine whether an implicitly given linear differential curve is unirational, and in the affirmative case, to compute a proper differential rational parametrization for the differential curve.

IMPLICITIZATION OF RATIONAL CURVES AND POLYNOMIAL SURFACES

In this paper, we first present a method for finding the implicit equation of the curve given by rational parametric equations. The method is based on the computation of <TEX>$Gr\"{o}bner$</TEX> bases. Then, another method for implicitization of curve and surface is given. In the case of rational curves, the method proceeds via giving the implicit polynomial f with indeterminate coefficients, substituting the rational expressions for the given curve and surface into the implicit polynomial to yield a rational expression <TEX>$\frac{g}{h}$</TEX> in the parameters. Equating coefficients of g in terms of parameters to 0 to get a system of linear equations in the indeterminate coefficients of polynomial f, and finally solving the linear system, we get all the coefficients of f, and thus we obtain the corresponding implicit equation. In the case of polynomial surfaces, we can similarly as in the case of rational curves obtain its implicit equation. This method is based on characteristic set theory. Some examples will show that our methods are efficient.

Bernstein–Bezoutian matrices and curve implicitization

A new application of Bernstein–Bezoutian matrices, a type of resultant matrices constructed when the polynomials are given in the Bernstein basis, is presented. In particular, the approach to curve implicitization through Sylvester and Bézout resultant matrices and bivariate interpolation in the usual power basis is extended to the case in which the polynomials appearing in the rational parametric equations of the curve are expressed in the Bernstein basis, avoiding the basis conversion from the Bernstein to the power basis. The coefficients of the implicit equation are computed in the bivariate tensor-product Bernstein basis, and their computation involves the bidiagonal factorization of the inverses of certain totally positive matrices.

The Proper Parametrization of a Special Class of Rational Parametric Equations

We give a proper reparametrization theorem for a set of rational parametric equations which is proper for all but one of its parameters. We also give an algorithm to determine whether a set of rational parametric equations belongs to this class, and if it does, we reparametrize it such that the new parametric equations are proper.

Implicitization of Partial Differential Rational Parametric Equations

In this paper, we propose algorithms for the following problems in the implicitization of a set of partial differential rational parametric equations P. (1)To find a characteristic set for the implicit prime ideal of P; (2) To find a canonical representation for the image of P; (3)To decide whether the parameters of P are independent, and if not, to re-parameterize P so that the new parametric equations have independent parameters; (4) To compute the inversion maps of P, and as a consequence, to decide whether P is proper.

A simple method for implicitizing rational curves and surfaces

This paper presents a simple method for converting rational parametric equations of curves and surfaces into implicit equations. The method proceeds via writing out the implicit polynomial F of estimated degree with indeterminate coefficients u i , substituting the rational expressions for the given parametric curve or surface into F to yield a rational expression g/ h in the parameter s (or s and t), equating the coefficients of g in terms of s (and t) to 0 to generate a sparse, partially triangular system of linear equations in u i with constant coefficients, and finally solving the linear system for u i . If a nontrivial solution is found, then an implicit polynomial is obtained; otherwise, one repeats the same process, increasing the degree of F. Our experiments show that this simple method is efficient. It performs particularly well in the presence of base points and may detect the dependency of parameters incidentally.

Implicitization of differential rational parametric equations

In this paper, we propose algorithms for the following problems in the implicitization of a set of differential rational parametric equations P. (1) To find a characteristic set for the implicit prime ideal of P. (2) To find a canonical representation for the image of P. (3) To decide whether the parameters of P are independent, and if not, to re-parameterize P so that the new parametric equations have independent parameters. (4) To compute the inversion maps of P, and as a consequence, to decide whether P is proper. If the implicit variety is of dimension one and P is not proper, to find a proper re-parameterization for P.

Implicitization of rational surfaces by means of polynomial interpolation

A method for finding the implicit equation of a surface given by rational parametric equations is presented. The method is based on an efficient computation of the resultant by means of classical multivariate polynomial interpolation. The used approach considerably reduces the problem of intermediate expression swell and it can easily be implemented in parallel.

Implicitization of rational parametric equations

Based on the Gröbner basis method, we present algorithms for a complete solution to the following problems in the implicitization of a set of rational parametric equations. (1) Find a basis of the implicit prime ideal determined by a set of rational parametric equations. (2) Decide whether the parameters of a set of rational parametric equations are independent. (3) If the parameters of a set of rational parametric equations are not independent, reparameterize the parametric equations so that the new parametric equations have independent parameters. (4) Compute the inversion maps of parametric equations, and as a consequence, give a method to decide whether a set of parametric equations is proper. (5) In the case of algebraic curves, find a proper reparameterization for a set of improper parametric equations.

On the parameterization of algebraic curves

In this paper, by using the concept of resolvents of a prime ideal introduced by Ritt, we give methods for constructing a hypersurface which is birational to a given irreducible variety and birational transformations between the hypersurface and the variety. In the case of algebraic curves, this implies that for an irreducible algebraic curveC, we can construct a plane curve which is birational toC. We also present a method to find rational parametric equations for a plane curve if it exists. Hence we have a complete method of parameterization for rational algebraic curves.

Rational hypersurface display

Algorithms are presented for polygonalizing implicitly defined, quadric and cubic hypersurfaces in n &amp;ge; 3 dimensional space and furthermore displaying their projections in 3D. The method relies on initially constructing the rational parametric equations of the implicitly defined hypersurfaces, and then polygonalizing these hypersurfaces by an adaptive generalized curvature dependent scheme. The number of hyperpolygons used are optimal, in that they are the order of the minimum number required for a smooth Gouraud like shading of the hypersurfaces. Such hypersurface projection displays should prove useful in scientific visualization applications. The curvature dependent polygonal meshes produced, should also prove very useful in finite difference and finite element analysis programs for multi-dimensional domains.

Automatic parameterization of rational curves and surfaces IV: algebraic space curves

For an irreducible algebraic space curve C that is implicitly defined as the intersection of two algebraic surfaces, f ( x , y , z ) = 0 and g ( x , y , z ) = 0, there always exists a birational correspondence between the points of C and the points of an irreducible plane curve P , whose genus is the same as that of C . Thus C is rational if the genus of P is zero. Given an irreducible space curve C = ( f ∩ g ), with f and g not tangent along C , we present a method of obtaining a projected irreducible plane curve P together with birational maps between the points of P and C . Together with [4], this method yields an algorithm to compute the genus of C , and if the genus is zero, the rational parametric equations for C . As a biproduct, this method also yields the implicit and parametric equations of a rational surface S containing the space curve C . The birational mappings of implicitly defined space curves find numerous applications in geometric modeling and computer graphics since they provide an efficient way of manipulating curves in space by processing curves in the plane. Additionally, having rational surfaces containing C yields a simple way of generating related families of rational space curves.

Automatic parameterization of rational curves and surfaces III: Algebraic plane curves

We present algorithms to compute the genus and rational parametric equations, for implicitly defined irreducible plane algebraic curves of arbitrary degree. Rational parameterizations exist for all irreducible algebraic curves of genus 0. The genus is computed by a complete analysis of the singularities of plane algebraic curves, using affine quadratic transformations. The rational parameterization techniques, essentially, reduce to solving symbolically systems of homogeneous linear equations and the computation of polynomial resultants.

Face area evaluation algorithm for solids

A scan curve algorithm is presented for evaluating the face area of solids in constructive solid geometry (CSG) representation. The applicable domain is limited to solids bounded by three classes of surfaces: all quadric surfaces, cylindrical surfaces, and surfaces of revolution where the last two have rational parametric equations for their generating curves. The algorithm has been implemented in Fortran 77 on a VAX 11/780 machine. The extensions of this algorithm may also be applied to the solution of the following three problems: boundary representation (B-rep) evaluation from CSG, face area evaluation for solids in B-rep, and triangulation of the faces of solids in CSG or B-rep.

Automatic parametrization of rational curves and surfaces II: cubics and cubicoids

Algorithms that can obtain rational and special parametric equations for degree three algebraic curves (cubics) and degree three algebraic surfaces (cubicoids), given their implicit equations are described. These algorithms have been implemented on a VAX8600 using VAXIMA.

Automatic parameretization of rational curves and surfaces 1: conics and conicoids

Given the implicit equation for degree two curves (conics) and degree two surfaces (conicoids), algorithms are described here, which obtain their corresponding rational parametric equations (a polynomial divided by another). These rational parameterizations are considered over the fields of rationals, reals and complex numbers. In doing so, solutions are given to important subproblems of finding rational and real points on the given conic curve or conicoid surface. Further polynomial parameterizations are obtained whenever they exist for the conics or conicoids. These algorithms have been implemented on a VAX-780 using VAXIMA.