﻿ 基于轴线光滑拼接的轴线异面管道拼接技术

# 基于轴线光滑拼接的轴线异面管道拼接技术Blending Technology of Tubes with Non-Coplanar Axes Based on Smooth Blending Axis

Abstract: The problem of smooth blending axes of non-coplanar tubes with cubic uniform B-spline curve is discussed. By the constraint of the control polygon, let B-spline curve pass through the vertices of the control polygon, and the curve is tangent to the edge of the control polygon. In this way, a piecewise cubic uniform B-spline curve can be constructed. Thus, this curve will be smoothly blended between axes of two non-coplanar tubes. And then, a smooth blending circular tube that takes B-spline curve as its axis is obtained.

1. 引言

$r\left(s\right)=\underset{i=0}{\overset{3}{\sum }}{B}_{i,3}\left(s\right){V}_{i}$

$\left\{\begin{array}{l}{N}_{0,3}=\frac{1}{6}\left(-{s}^{3}+3{s}^{2}-3s+1\right),\\ {N}_{1,3}=\frac{1}{6}\left(3{s}^{3}-6{s}^{2}+4\right),\\ {N}_{2,3}=\frac{1}{6}\left(-3{s}^{3}+3{s}^{2}+3s+1\right),\\ {N}_{3,3}=\frac{1}{6}{s}^{3}.\end{array}$

2. 构造基于轴线光滑拼接的轴线异面管道的光滑拼接管道

${\Phi }_{1}:\left\{\begin{array}{l}x={x}_{1}+{a}_{\text{1}}{N}_{11}\mathrm{cos}\phi +{a}_{\text{1}}{B}_{11}\mathrm{sin}\phi ,\\ y={y}_{1}+{b}_{1}s+{a}_{\text{1}}{N}_{12}\mathrm{cos}\phi +{a}_{\text{1}}{B}_{12}\mathrm{sin}\phi ,\\ z={a}_{\text{1}}{N}_{13}\mathrm{cos}\phi +{a}_{\text{1}}{B}_{13}\mathrm{sin}\phi .\end{array}$${\Phi }_{2}:\left\{\begin{array}{l}x={a}_{\text{2}}{N}_{21}\mathrm{cos}\phi +{a}_{\text{2}}{B}_{21}\mathrm{sin}\phi ,\\ y={y}_{2}+{a}_{\text{2}}{N}_{22}\mathrm{cos}\phi +{a}_{\text{2}}{B}_{22}\mathrm{sin}\phi ,\\ z={z}_{2}+{c}_{2}s+{a}_{\text{2}}{N}_{23}\mathrm{cos}\phi +{a}_{\text{2}}{B}_{23}\mathrm{sin}\phi .\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\phi \in \left[0,2\text{π}\right]$

${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}+{b}_{\text{2}}\cdot s,\\ z={z}_{2}+{c}_{2}s.\end{array}$

${V}_{0}\left({x}_{0},{y}_{0},0\right),{V}_{1}\left({x}_{1},{y}_{1},0\right)$${L}_{1}$ 上的两个点， ${V}_{2}\left(0,{y}_{2},{z}_{2}\right),{V}_{1}\left(0,{y}_{3},{z}_{3}\right)$${L}_{2}$ 上的两个点，在 ${V}_{0}{V}_{1}$ 的反向延长线上取 ${V}_{-1}\left({x}_{1},{y}_{-1},0\right)$ ，使得 $|{V}_{1}-{V}_{0}|=|{V}_{0}-{V}_{-1}|$ ，在 ${V}_{2}{V}_{3}$ 的延长线上取 ${V}_{4}$ ，使 $|{V}_{2}-{V}_{3}|=|{V}_{3}-{V}_{4}|$ 。则以 ${V}_{1},{V}_{0},{V}_{1},{V}_{2},{V}_{3},{V}_{4}$ 为特征多边形的顶点的均匀三次B样条曲线通过顶点 ${V}_{0},{V}_{3}$ ，且与特征多边形的第一条边 ${V}_{0}{V}_{1}$ 与第三条边 ${V}_{2}{V}_{3}$ 相切。

3. 构造与轴线异面的圆管道Φ1与Φ2光滑拼接的三段光滑圆管道

$p\left(s,\phi \right)=\left\{\begin{array}{l}x\left(s\right)+a{N}_{1}\left(s\right)\mathrm{cos}\phi +a{B}_{1}\left(s\right)\mathrm{sin}\phi ,\\ y\left(s\right)+a{N}_{2}\left(s\right)\mathrm{cos}\phi +a{B}_{2}\left(s\right)\mathrm{sin}\phi ,\\ z\left(s\right)+a{N}_{3}\left(s\right)\mathrm{cos}\phi +a{B}_{3}\left(s\right)\mathrm{sin}\phi .\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}s\in \left[0,1\right],\phi \in \left(0,\text{π}\right)$

$N=\left({N}_{1}\left(s\right),{N}_{2}\left(s\right),{N}_{3}\left(s\right)\right),B=\left({B}_{1}\left(s\right),{B}_{2}\left(s\right),{B}_{3}\left(s\right)\right)$

${a}_{1}\ne {a}_{2}$ 时，再构造一个光滑拼接两个轴线异面的管道的某两个母线 ${r}_{\text{1}}\left(s\right)$${r}_{\text{2}}\left(s\right)$ 的B样条曲线 ${r}^{\prime }\left(s\right)$ ，则光滑拼接粗细不同的轴线异面的管道的拼接管道的表示式为

$p\left(s,\phi \right)=\left\{\begin{array}{l}x\left(s\right)+d\left(s\right){N}_{1}\left(s\right)\mathrm{cos}\phi +d\left(s\right){B}_{1}\left(s\right)\mathrm{sin}\phi ,\\ y\left(s\right)+d\left(s\right){N}_{2}\left(s\right)\mathrm{cos}\phi +d\left(s\right){B}_{2}\left(s\right)\mathrm{sin}\phi ,\\ z\left(s\right)+d\left(s\right){N}_{3}\left(s\right)\mathrm{cos}\phi +d\left(s\right){B}_{3}\left(s\right)\mathrm{sin}\phi .\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}s\in \left[0,1\right],\phi \in \left(0,\text{π}\right)$

4. 拼接实例

Figure 1. Three-segment G1-blending diagram is same radiuses tubes whose axes are in non-coplaner take the B-spline curve as its axis

Figure 2. Three-segment G1-blending diagram is different radiuses tubes whose axes are in non-coplaner take the B-spline curve as its axis

Figure 3. Comparison of the smoothness of the same control vertex B-spline curve and Bézier curve

5. 结束语

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