﻿ 有理Bézier曲线及其在粗细不同管道拼接中的应用 Rational Bézier Curve and Its Application in Blending of Thickness Different Tubes

Vol. 07  No. 09 ( 2018 ), Article ID: 26703 , 6 pages
10.12677/AAM.2018.79130

Rational Bézier Curve and Its Application in Blending of Thickness Different Tubes

Taotuge1, Wurengaowa1, Fang Wang2, Genzhu Bai3*

1Inner Mongolia Tongliao Vocational College, Tongliao Inner Mongolia

2Zhejiang Changzheng Vocational & Technical College, Hangzhou Zhejiang

3Inner Mongolia University for the Nationalities, Tongliao Inner Mongolia

Received: Aug. 13th, 2018; accepted: Aug. 29th, 2018; published: Sep. 5th, 2018

ABSTRACT

In this paper, the rational Bézier curve can be adjusted and controlled in the shape of a curve near a particular control point. Then, on the basis of the smooth blending circular tubes with the two different radiuses, the axes of the tubes are non-coplanar lines. The blending tubes that meet the conditions of G0 and G1 are constructed separately. A new tubes blending is obtained. It has theoretical significance and application value.

Keywords:Shape Parameters, Rational Bézier Curve, Axis, Tubes, Smooth Blending

1内蒙古通辽职业学院，内蒙古 通辽

2浙江长征职业技术学院，浙江 杭州

3内蒙古民族大学，内蒙古 通辽

1. 引言

${\Phi }_{1}:\left\{\begin{array}{l}x={X}_{1}+a\mathrm{cos}\phi +a\mathrm{sin}\phi ,\\ y={Y}_{1}+{B}_{1}s+a\mathrm{cos}\phi +a\mathrm{sin}\phi ,\\ z=a\mathrm{cos}\phi +a\mathrm{sin}\phi .\end{array}$ (1)

${L}_{1}:\left\{\begin{array}{l}x={X}_{1}+0\cdot s,\\ y={Y}_{1}+{B}_{1}s,\\ z=0+0\cdot s,\end{array}$${L}_{2}:\left\{\begin{array}{l}x=0+0\cdot s,\\ y={Y}_{2}+0\cdot s,\\ z={Z}_{2}+{C}_{2}s.\end{array}$ (2)

$r\left(t\right)=\frac{\underset{i=0}{\overset{n}{\sum }}{B}_{n,i}\left(t\right){w}_{i}{V}_{i}}{\underset{i=0}{\overset{n}{\sum }}{B}_{n,i}\left(t\right){w}_{i}}$

n为次Bézier曲线和n次有理Bézier曲线。其中 ${B}_{n,i}\left(t\right),i=0,1,\cdots ,n$ 为Bernstein基函数。 ${w}_{i}\ne 0,i=0,1,\cdots ,n$ 为对应控制顶点的权因子。有理Bézier曲线和Bézier曲线一样通过首、末顶点并和特征多边形的首、末两条边相切。

(3)

$\Phi :\left\{\begin{array}{l}x=\underset{i=0}{\overset{3}{\sum }}{B}_{3,i}\left(s\right){x}_{i}+r{N}_{1}\left(s\right)\mathrm{cos}\phi +r{B}_{1}\left(s\right)\mathrm{sin}\phi ,\\ y=\underset{i=0}{\overset{3}{\sum }}{B}_{3,i}\left(s\right){y}_{i}+r{N}_{2}\left(s\right)\mathrm{cos}\phi +r{B}_{2}\left(s\right)\mathrm{sin}\phi ,\\ z=\underset{i=0}{\overset{3}{\sum }}{B}_{3,i}\left(s\right){z}_{i}+r{N}_{3}\left(s\right)\mathrm{cos}\phi +r{B}_{3}\left(s\right)\mathrm{sin}\phi .\end{array}\text{}s\in \left[0,1\right],\text{}\phi \in \left(0,\text{π}\right).$ (4)

$p\left(s,\phi \right)=\left\{\begin{array}{l}x\left(s\right)+a{N}_{1}\left(s\right)\mathrm{cos}\phi +a{B}_{1}\left(s\right)\mathrm{sin}\phi ,\\ y\left(s\right)+a{N}_{2}\left(s\right)\mathrm{cos}\phi +a{B}_{2}\left(s\right)\mathrm{sin}\phi ,s\in \left[0,1\right],\text{}\phi \in \left(0,\text{π}\right)\\ z\left(s\right)+a{N}_{3}\left(s\right)\mathrm{cos}\phi +a{B}_{3}\left(s\right)\mathrm{sin}\phi .\end{array}$

2. 构造粗细不同的轴线异面管道的G0-拼接管道

1) 存在光滑拼接两个管道的轴线 ${\gamma }_{1}\left(s\right)$${\gamma }_{2}\left(s\right)$ 的有理Bézier曲线 $\gamma \left(s\right)$ ，即

${\gamma }_{1}\left(1\right)=\gamma \left(0\right),\gamma \left(1\right)={\gamma }_{2}\left(0\right).$

2) 在轴线光滑拼接点与轴线垂直的平面处的管道半径相同，即

$p\left(s,\phi \right)=\left\{\begin{array}{l}x\left(s\right)+\alpha \left(s\right){N}_{1}\left(s\right)\mathrm{cos}\phi +\alpha \left(s\right){B}_{1}\left(s\right)\mathrm{sin}\phi ,\\ y\left(s\right)+\alpha \left(s\right){N}_{2}\left(s\right)\mathrm{cos}\phi +\alpha \left(s\right){B}_{2}\left(s\right)\mathrm{sin}\phi ,s\in \left[0,1\right],\text{}\phi \in \left(0,\text{π}\right).\\ z\left(s\right)+\alpha \left(s\right){N}_{3}\left(s\right)\mathrm{cos}\phi +\alpha \left(s\right){B}_{3}\left(s\right)\mathrm{sin}\phi .\end{array}$

3. 构造光滑拼接粗细不同的轴线异面管道的G1-拼接管道

1) 存在光滑拼接两个轴线异面管道的轴线 ${\gamma }_{1}\left(u\right)$${\gamma }_{2}\left(u\right)$ 的有理Bézier曲线 $\gamma \left(u\right)$ ，使得 ${\gamma }_{1}\left(1\right)=\gamma \left(0\right),\gamma \left(1\right)={\gamma }_{2}\left(0\right).$

2) 存在光滑拼接两个轴线异面管道的母线 ${{\gamma }^{\prime }}_{1}\left(u\right)$${{\gamma }^{\prime }}_{2}\left(u\right)$ 的有理Bézier曲线 ${\gamma }^{\prime }\left(u\right)$ ，使得 ${{\gamma }^{\prime }}_{1}\left(1\right)={\gamma }^{\prime }\left(0\right),{\gamma }^{\prime }\left(1\right)={{\gamma }^{\prime }}_{2}\left(0\right)$

$p\left(s,\phi \right)=\left\{\begin{array}{l}x\left(s\right)+d\left(s\right){N}_{1}\left(s\right)\mathrm{cos}\phi +d\left(s\right){B}_{1}\left(s\right)\mathrm{sin}\phi ,\\ y\left(s\right)+d\left(s\right){N}_{2}\left(s\right)\mathrm{cos}\phi +d\left(s\right){B}_{2}\left(s\right)\mathrm{sin}\phi ,s\in \left[0,1\right],\text{}\phi \in \left(0,\text{π}\right).\\ z\left(s\right)+d\left(s\right){N}_{3}\left(s\right)\mathrm{cos}\phi +d\left(s\right)B\left(s\right)\mathrm{sin}\phi .\end{array}$ (5)

Figure 1. Blending graph G0 with rational Bézier curve as axis

Figure 2. Blending graph G1 with rational Bézier curve as axis

4. 结束语

Rational Bézier Curve and Its Application in Blending of Thickness Different Tubes[J]. 应用数学进展, 2018, 07(09): 1127-1132. https://doi.org/10.12677/AAM.2018.79130

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11. NOTES

*通讯作者。