Higher Order Derivatives of a Rational Bézier Curve

1995 ◽  
Vol 57 (3) ◽  
pp. 246-253 ◽  
Author(s):  
Guo-Zhao Wang ◽  
Guo-Jin Wang
Keyword(s):  
Higher Order ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Rational Bézier Curve ◽  
Derivatives Of

scholarly journals Designing Developable C-Bézier Surface with Shape Parameters

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 402 ◽  
Author(s):  
Caiyun Li ◽  
Chungang Zhu
Keyword(s):  
Architectural Design ◽  
Basis Functions ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Developable Surface ◽  
Rational Bézier Curves ◽  
Polynomial Representations ◽  
Rational Bézier Curve ◽  
Shape Controlling ◽  
Two Parameters

Developable surface plays an important role in geometric design, architectural design, and manufacturing of material. Bézier curve and surface are the main tools in the modeling of curve and surface. Since polynomial representations can not express conics exactly and have few shape handles, one may want to use rational Bézier curves and surfaces whose weights control the shape. If we vary a weight of rational Bézier curve or surface, then all of the rational basis functions will be changed. The derivation and integration of the rational curve will yield a high degree curve, which means that the shape of rational Bézier curve and surface is not easy to control. To solve this problem of shape controlling for a developable surface, we construct C-Bézier developable surfaces with some parameters using a dual geometric method. This yields properties similar to Bézier surfaces so that it is easy to design. Since C-Bézier basis functions have only two parameters in every basis, we can control the shape of the surface locally. Moreover, we derive the conditions for C-Bézier developable surface interpolating a geodesic.

A new fractional rational Bézier curve

2005 ◽  
Vol 9 (3) ◽  
pp. 216-218
Author(s):  
Bei-bei Wu ◽  
Chuan-qing GU
Keyword(s):  
Bézier Curve ◽  
Bezier Curve ◽  
Rational Bézier Curve

A new bound on the magnitude of the derivative of rational Bézier curve

2013 ◽  
Vol 30 (2) ◽  
pp. 175-180 ◽  
Author(s):  
Chongyang Deng ◽  
Yajuan Li
Keyword(s):  
Bézier Curve ◽  
Bezier Curve ◽  
Rational Bézier Curve

scholarly journals Rational Bézier Curve and Its Application

2017 ◽  
Vol 06 (08) ◽  
pp. 935-941 ◽  
Author(s):  
芳 王
Keyword(s):  
Bézier Curve ◽  
Bezier Curve ◽  
Rational Bézier Curve

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

2021 ◽  
Vol 2106 (1) ◽  
pp. 012017
Author(s):  
Y R Anwar ◽  
H Tasman ◽  
N Hariadi
Keyword(s):  
Gröbner Basis ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Conic Section ◽  
Control Points ◽  
Grobner Basis ◽  
Parametric Curve ◽  
Wide Range ◽  
Rational Bézier Curve ◽  
Implicit Polynomial

Abstract 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.

Sign in / Sign up

Export Citation Format

Share Document