﻿ 关于圆锥曲面的光滑拼接问题研究

# 关于圆锥曲面的光滑拼接问题研究Research on Smooth Conic Surface Splicing Problem

Abstract: Based on the research results on multivariate though laser interpolation problem of algebraic surface and space curve, this paper mainly studies the stitching problems of the along conical surface, gets a set of decomposition methods of quadratic smoothness on conic surfaces by using Lagrange interpolation method, and obtains a set of quadratic splicing polynomials satisfying the smooth splicing along conic surfaces, to simplify the surface of stitching process. In this paper, we use the experimental example of conic surface (as shown in Figure 1) to implement the method presented in this paper, and verify the effectiveness of the method.

1. 引言

2. 基本定义和主要定理

${P}_{n}^{\left(3\right)}=\left\{\underset{0\le i+j+k\le n}{\sum }{a}_{ijk}{x}^{i}{y}^{j}{z}^{k}|{a}_{ijk}\in R\right\}$

${d}_{n}\left(2\right)=\left(\begin{array}{c}n+3\\ 3\end{array}\right)-\left(\begin{array}{c}n+1\\ 3\end{array}\right)=\left\{\begin{array}{l}\frac{1}{6}\left(n+3\right)\left(n+2\right)\left(n+1\right)\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{ }n<\text{2}\\ \frac{1}{6}k\left[3n\left(n-\text{2}\right)+12n+{\text{2}}^{2}-1\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ge \text{2}\end{array}$ (1)

$A={\left\{{Q}_{i}\right\}}_{i=1}^{{d}_{n}\left(2\right)}$ 是不可约代数曲线 $c\left(x,y,z\right)=\text{0}$ 上的 ${d}_{n}\left(2\right)$ 个互不相同的点，给定任意一组实数组 ${\left\{{f}_{i}\right\}}_{i=1}^{{d}_{n}\left(2\right)}$，并且能够找到关于该实数组的一个多项式 $g\left(x,y,z\right)\in {P}_{n}^{\left(3\right)}$，使得所求 $g\left(x\right)$ 满足下述条件：

$g\left({Q}_{i}\right)={f}_{i},i=1,\cdots ,n;i=1,\cdots ,{d}_{n}\left(k\right).$ (2)

$g\left({Q}_{i}\right)=\text{0},i=1,\cdots ,{d}_{n}\left(2\right).$

$g\left(X\right)=c\left(X\right)r\left( X \right)$

$g\left(X\right)=c\left(X\right)r\left( X \right)$

$g\left(X\right)=c\left(X\right)r\left(X\right)+d\left( X \right)$

$g\left({Q}_{i}\right)=c\left({Q}_{i}\right)r\left({Q}_{i}\right)+d\left({Q}_{i}\right)=d\left({Q}_{i}\right)\ne 0$$g\left({Q}_{i}\right)=\text{0,}\left(i=1,\cdots ,{d}_{n}\left(2\right)\right)$ 矛盾。

$g\left(X\right)=c\left(X\right)r\left(X\right)$

3. 实验算例

${Q}_{1}\left(-\text{24},-7,\text{0}\right)$${Q}_{2}\left(-20,-15,\text{0}\right)$${Q}_{3}\left(-7,-24,\text{0}\right)$

${Q}_{4}\left(7,24,\text{0}\right)$${Q}_{5}\left(7,-24,\text{0}\right)$${Q}_{6}\left(-7,24,\text{0}\right)$

${Q}_{7}\left(15,20,\text{0}\right)$${Q}_{8}\left(-15,20,\text{0}\right)$${Q}_{9}\left(15,-20,\text{0}\right)$

$z=0$ 时，这九个点恰好在在曲线 $c\left(X\right):{x}^{2}+{y}^{2}-\text{2}{5}^{2}=0$ 上，由定义知， $\left\{{Q}_{1},\cdots ,{Q}_{9}\right\}$ 构成了定义在相交2次代数曲线 $c\left(X\right):{x}^{2}+{y}^{2}-\text{2}{5}^{2}=0$ 的一组光滑拼接点组。

$g\left(x,y,z\right)={a}_{1}{x}^{2}+{a}_{2}{y}^{2}+{a}_{3}{z}^{2}+{a}_{4}xy+{a}_{5}xz+{a}_{6}yz+{a}_{7}x+{a}_{8}y+{a}_{9}z+{a}_{\text{10}}$

$A=\left[\begin{array}{cccccccccc}{\left(-24\right)}^{2}& {\left(-7\right)}^{2}& 0& 168& 0& 0& -24& -7& 0& 1\\ {\left(-20\right)}^{2}& {\left(-15\right)}^{2}& 0& 300& 0& 0& -20& -15& 0& 1\\ {\left(-7\right)}^{2}& {\left(-24\right)}^{2}& 0& 168& 0& 0& -7& -24& 0& 1\\ {7}^{2}& {\left(-24\right)}^{2}& 0& -168& 0& 0& 7& -24& 0& 1\\ {7}^{2}& {24}^{2}& 0& 168& 0& 0& 7& 24& 0& 1\\ {\left(-7\right)}^{2}& {24}^{2}& 0& -168& 0& 0& -7& 24& 0& 1\\ {15}^{2}& {20}^{2}& 0& 300& 0& 0& 15& 20& 0& 1\\ {\left(-15\right)}^{2}& {20}^{2}& 0& -300& 0& 0& -15& 20& 0& 1\\ {15}^{2}& {\left(-20\right)}^{2}& 0& -300& 0& 0& 15& -20& 0& 1\end{array}\right]$

$X=\left[\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\\ {a}_{4}\\ {a}_{5}\\ {a}_{6}\\ {a}_{7}\\ {a}_{8}\\ {a}_{9}\\ {a}_{10}\end{array}\right]$$B=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right]$，此时解得 $\left\{\begin{array}{l}{a}_{1}=1\hfill \\ {a}_{2}=1\hfill \\ {a}_{3}=0\hfill \\ {a}_{4}=0\hfill \\ {a}_{5}=0\hfill \\ {a}_{6}=0\hfill \\ {a}_{7}=0\hfill \\ {a}_{8}=0\hfill \\ {a}_{9}=0\hfill \\ {a}_{10}=-{25}^{2}\hfill \end{array}$

$g\left(x,y,z\right)={x}^{2}+{y}^{2}-{25}^{2}=\text{0}$

Figure 1. Effect diagram of smooth conic surface Mosaic

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