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[x1,...,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 P₀(x0, y0), P1(x1, y1), P2(x2, y2) in R²and weights ω0; ω1; ω2, where the weights ωi 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 ω0 = ω2 = 1 and ω1 = ω for any control points P0(x0, y0), P1(x1, y1), and P2(x2, y2). 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 P0(x0, y0), P1(x1, y1), P2(x2, y2) 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.