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.

Polynomial Fitting Algorithm Based on Neural Network

As a method of function approximation, polynomial fitting has always been the main research hotspot in mathematical modeling. In many disciplines such as computer, physics, biology, neural networks have been widely used, and most of the applications have been transformed into fitting problems using neural networks. One of the main reasons that neural networks can be widely used is that it has a certain sense of universal approximation. In order to fit the polynomial, this paper constructs a three-layer feedforward neural network, uses Taylor series as the activation function, and determines the number of hidden layer neurons according to the order of the polynomial and the dimensions of the input variables. For explicit polynomial fitting, this paper uses non-linear functions as the objective function, and compares the fitting effects under different orders of polynomials. For the fitting of implicit polynomial curves, the current popular polynomial fitting algorithms are compared and analyzed. Experiments have proved that the algorithm used in this paper is suitable for both explicit polynomial fitting and implicit polynomial fitting. The algorithm is relatively simple, practical, easy to calculate, and can efficiently achieve the fitting goal. At the same time, the computational complexity is relatively low, which has certain application value.

On a New Distance Approximation for an Implicit Polynomial Manifold Fitting

The application of a new approximate point-to-algebraic manifold distance formula is suggested to the geometric approach to curve fitting and surface reconstruction using implicit polynomial manifolds. A brief overview of the fitting methods features for implicit algebraic manifolds is given. To illustrate the possibilities of a new approximate point-to-manifold distance formula, the equidistant curves of the exact distance, Samson’s distance and the present formula are given. A four-step algorithm for implicit algebraic manifold fitting is proposed, using one of the algebraic fitting methods at the initial step, the present approximate formula for the distance finding to calculate the geometric criterion of approximation quality and an optimization method for updating the value of the vector of coefficients of the manifold. The first results of the proposed algorithm on test data are briefly characterized. In conclusion, the tasks and directions for further research are described.