TY - JOUR

T1 - Determining implicit equation of conic section from quadratic rational Bézier curve using Gröbner basis

AU - Anwar, Y. R.

AU - Tasman, H.

AU - Hariadi, N.

N1 - Publisher Copyright:
© 2021 Institute of Physics Publishing. All rights reserved.

PY - 2021/11/23

Y1 - 2021/11/23

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85121439169&partnerID=8YFLogxK

U2 - 10.1088/1742-6596/2106/1/012017

DO - 10.1088/1742-6596/2106/1/012017

M3 - Conference article

AN - SCOPUS:85121439169

SN - 1742-6588

VL - 2106

JO - Journal of Physics: Conference Series

JF - Journal of Physics: Conference Series

IS - 1

M1 - 012017

T2 - International Conference on Mathematical and Statistical Sciences 2021, ICMSS 2021

Y2 - 15 September 2021 through 16 September 2021

ER -