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.

Model Matematika Kurva Kuartik Bezier Hasil Modifikasi Kubik Bezier

The creative industries have become the government's attention for contributing to economic accretion. But due the lack of artistic creativity and appeal, the evolution of creative industries craft section is not optimal. So that it was needed a variations of relief items to increase the attractiveness. In general, industrial objects design are still limited to the space geometry objects or a Bezier curve of degree two. Therefore, Bezier curves of degree is selected and modified it into a quartic Bezier forms and then applied to the design of industrial objects (glassware). The purpose of this research is to determine the formula of quartic Bezier from of qubic Bezier modifications and to determine the rotary surface shape of quartic Bezier from cubic Bezier modifications. Then, from some form of revolving surface of modified cubic Bezier the glassware designs are generated. The results of this research are, first, the formula of quartic Bezier result of Bezier cubic modifications. Second, the form of revolving surface of modified cubic Bezier which is influenced by five control points and parameter selection . For further Research it is expected to develop a modification of cubic Bezier into Bezier of degree-

Airfoil Shape and Angle of Attack Optimization Based On Bézier Curve and Multi-Island Genetic Algorithm

In order to improve the aerodynamic performance of the airfoil, the airfoil shape and the angle of attack (AOA) are optimized at the same time by the Multi-island Genetic Algorithm in this paper. The goal of the optimization is to maximize the lift-to-drag ratio which is calculated by computational fluid dynamics method. The airfoil is parameterized by the Bézier curve. The thickness and the camber of airfoil are no longer restricted to ensure a wide range of airfoil generation. The airfoil is optimized under different Reynolds numbers. The optimized airfoils obtained by the unconstrained AOA method are compared with several standard airfoils. The results show that the maximum lift-to-drag ratio of the optimized airfoil is much greater than the compared airfoils, and the optimized airfoils have good aerodynamic characteristics in a wide range of angle of attack. By comparing with the optimized airfoils obtained by the constrained AOA method, it shows that the constrained AOA method can't guarantee that the pre-constrained angle of attack is the optimal angle of attack of the airfoil, nor can obtain the maximum lift-to-drag ratio airfoil of all angles of attack and all airfoils. However, by using the angle of attack as one of the optimization variables, these problems can be solved well.