Constraint Equations for a Planar Parallel Platform

<p>The aim of this thesis is to apply the Grünwald–Blaschke kinematic mapping to standard types of parallel general planar three-legged platforms in order to obtain the univariate polynomials which provide the solution of the forward kinematic problem. We rely on the method of Gröbner basis to reach these univariate polynomials. The Gröbner basis is determined from the constraint equations of the three legs of the platforms. The degrees of these polynomials are examined geometrically based on Bezout’s Theorem. The principle conclusion is that the univariate polynomials for the symmetric platforms under circular constraints are of degree six, which describe the maximum number of real solutions. The univariate polynomials for the symmetric platforms under linear constraints are of degree two, that describe the maximum number of real solutions.</p>

Constraint Equations for a Planar Parallel Platform

<p>The aim of this thesis is to apply the Grünwald–Blaschke kinematic mapping to standard types of parallel general planar three-legged platforms in order to obtain the univariate polynomials which provide the solution of the forward kinematic problem. We rely on the method of Gröbner basis to reach these univariate polynomials. The Gröbner basis is determined from the constraint equations of the three legs of the platforms. The degrees of these polynomials are examined geometrically based on Bezout’s Theorem. The principle conclusion is that the univariate polynomials for the symmetric platforms under circular constraints are of degree six, which describe the maximum number of real solutions. The univariate polynomials for the symmetric platforms under linear constraints are of degree two, that describe the maximum number of real solutions.</p>

Determining implicit equation of conic section from quadratic rational Bézier curve using Gröbner basis

The Gröbner Basis is a subset of finite generating polynomials in the ideal of the polynomial ring k[x 1,…,xn ]. The Gröbner basis has a wide range of applications in various areas of mathematics, including determining implicit polynomial equations. The quadratic rational Bézier curve is a rational parametric curve that is generated by three control points P 0(x 0,y 0), P 1(xi ,yi ), P 2(x 2,y 2) in ℝ2 and weights ω 0, ω 1, ω 2, where the weights ω i are corresponding to control points Pi (xi, yi ), for i = 0,1, 2. According to Cox et al (2007), the quadratic rational Bézier curve can represent conic sections, such as parabola, hyperbola, ellipse, and circle, by defining the weights ω 0 = ω 2 = 1 and ω 1 = ω for any control points P 0(x 0, y 0), P 1(x 1, y 1), and P 2(x 2, y 2). This research is aimed to obtain an implicit polynomial equation of the quadratic rational Bézier curve using the Gröbner basis. The polynomial coefficients of the conic section can be expressed in the term of control points P 0(x 0, y 0), P 1(x 1, y 1), P 2(x 2, y 2) and weight ω, using Wolfram Mathematica. This research also analyzes the effect of changes in weight ω on the shape of the conic section. It shows that parabola, hyperbola, and ellipse can be formed by defining ω = 1, ω > 1, and 0 < ω < 1, respectively.

An Algorithm to Compute the H-Bases for Ideals of Subalgebras

The concept of H-bases, introduced long ago by Macauly, has become an important ingredient for the treatment of various problems in computational algebra. The concept of H-bases is for ideals in polynomial rings, which allows an investigation of multivariate polynomial spaces degree by degree. Similarly, we have the analogue of H-bases for subalgebras, termed as SH-bases. In this paper, we present an analogue of H-bases for finitely generated ideals in a given subalgebra of a polynomial ring, and we call them “HSG-bases.” We present their connection to the SAGBI-Gröbner basis concept, characterize HSG-basis, and show how to construct them.

Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

In a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.