圆柱面方程与马氏距离

青青子衿

已知圆柱面的轴线方程\(\,\,{\large{l}}\colon\,\left\{\begin{split}
a_{11}x+a_{12}y+a_{13}z&=b_{1}\\
a_{21}x+a_{22}y+a_{23}z&=b_{2}\\
\end{split}\right.\)
则其准线圆半径为\(\,R\,\)的圆柱面方程为
\[\Large{\gamma\,\!P^2-2\beta\,\!PQ+\alpha\,\!Q^2=(\alpha\gamma-\beta^2)R^2}\]
其中
\begin{align*}
\alpha&={a_{\overset{}{11}}}\!\!^2+{a_{\overset{}{12}}}\!\!^2+{a_{\overset{}{13}}}\!\!^2\\
\beta&={a_{\overset{}{11}}}{a_{\overset{}{21}}}+{a_{\overset{}{12}}}{a_{\overset{}{22}}}+{a_{\overset{}{13}}}{a_{\overset{}{23}}}\\
\gamma&={a_{\overset{}{21}}}\!\!^2+{a_{\overset{}{22}}}\!\!^2+{a_{\overset{}{23}}}\!\!^2\\
\\
P&=a_{11}x+a_{12}y+a_{13}z-b_{1}\\
Q&=a_{21}x+a_{22}y+a_{23}z-b_{2}\\
\end{align*}

此外，如果记直线的系数矩阵为\(\boldsymbol{A}=\begin{pmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}
\end{pmatrix}=\begin{pmatrix}
\boldsymbol{n}_{1}\\
\boldsymbol{n}_{2}
\end{pmatrix}\)，常数列向量\(\boldsymbol{b}=\begin{pmatrix}
b_{1}\\
b_{2}
\end{pmatrix}\)，
曲面位置向量\(\boldsymbol{X}=\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}\)，故发现可以将圆柱面方程写成矩阵的形式\(\,\Big(\boldsymbol{A}\boldsymbol{X}-\boldsymbol{b}\Big)^{\mathrm{T}}\Big(\boldsymbol{A}\boldsymbol{A}^{\mathrm{T}}\Big)^{-1}\Big(\boldsymbol{A}\boldsymbol{X}-\boldsymbol{b}\Big)=R^2\,\)
感觉这个形式与多元统计分析里的马氏距离有些类似


已知圆柱面的轴线方程为\(\,\,{\large{l}}\colon\,\left\{\begin{split}
x+2y-5z&=3\\
2x+3y-6z&=7
\end{split}\right.\)，求其准线圆半径为3的圆柱面方程.

Ans:
17x^2+10y^2+25z^2+24xy+6xz-8yz-146x-100y-38z+81=0

Code:

2. Cross[{x - 5, y + 1, z}, {-3, 4, 1}].Cross[{x - 5, y + 1, z}, {-3, 4,
3.      1}] - 9*26 // Factor
4. -Det[({{30, 38, x + 2*y - 5*z - 3},
5.    {38, 49, 2*x + 3*y - 6*z - 7},
6.    {x + 2*y - 5*z - 3, 2*x + 3*y - 6*z - 7, 9}})]

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叶伯英,李艳敏.点到空间直线距离公式的矩阵表示[J].高等数学研究,2007,(第2期).

青青子衿

回复 1# 青青子衿

\begin{align*}
\Large{\begin{vmatrix}
\alpha&\beta&P\\
\beta&\gamma&Q\\
P&Q&R^2
\end{vmatrix}}&\>\>\Large{=}\>\>0\\
\\
\Large{\begin{vmatrix}
\boldsymbol{A}\boldsymbol{A}^{\mathrm{T}}&\boldsymbol{A}\boldsymbol{X}-\boldsymbol{b}\\
\Big(\boldsymbol{A}\boldsymbol{X}-\boldsymbol{b}\Big)^{\mathrm{T}}&R^2
\end{vmatrix}}&\>\>\Large{=}\>\>0\\
\\
\Large{\begin{vmatrix}
\langle\boldsymbol{n}_1,\boldsymbol{n}_1\rangle&\langle\boldsymbol{n}_1,\boldsymbol{n}_2\rangle&\langle\boldsymbol{n}_1,\boldsymbol{X}\rangle-b_1\\
\langle\boldsymbol{n}_2,\boldsymbol{n}_1\rangle&\langle\boldsymbol{n}_2,\boldsymbol{n}_2\rangle&\langle\boldsymbol{n}_2,\boldsymbol{X}\rangle-b_2\\
\langle\boldsymbol{n}_1,\boldsymbol{X}\rangle-b_1&\langle\boldsymbol{n}_2,\boldsymbol{X}\rangle-b_2&R^2
\end{vmatrix}}&\>\>\Large{=}\>\>0
\end{align*}

青青子衿

青青子衿 发表于 2020-8-22 20:24
回复 1# 青青子衿
\begin{align*}
\begin{vmatrix}
\langle\boldsymbol{n}_1,\boldsymbol{n}_1\rangle&\langle\boldsymbol{n}_1,\boldsymbol{n}_2\rangle&\langle\boldsymbol{n}_1,\boldsymbol{X}\rangle-b_1\\
\langle\boldsymbol{n}_2,\boldsymbol{n}_1\rangle&\langle\boldsymbol{n}_2,\boldsymbol{n}_2\rangle&\langle\boldsymbol{n}_2,\boldsymbol{X}\rangle-b_2\\
\langle\boldsymbol{n}_1,\boldsymbol{X}\rangle-b_1&\langle\boldsymbol{n}_2,\boldsymbol{X}\rangle-b_2&R^2
\end{vmatrix}&=0
\end{align*}

\begin{align*}
\left\Vert\dfrac{\begin{vmatrix}
\langle\boldsymbol{n}_1,\boldsymbol{X}\rangle-b_1&\langle\boldsymbol{n}_1,\boldsymbol{\boldsymbol{n}_2}\rangle\\
\langle\boldsymbol{n}_2,\boldsymbol{X}\rangle-b_2&\Vert\boldsymbol{n}_2\Vert^2
\end{vmatrix}}{\>\>\>\Vert\boldsymbol{n}_1\Vert^2\Vert\boldsymbol{n}_2\Vert^2-\langle\boldsymbol{n}_1,\boldsymbol{n}_2\rangle^2}\,\boldsymbol{n}_1
+\dfrac{
\begin{vmatrix}
\Vert\boldsymbol{n}_1\Vert^2&\langle\boldsymbol{n}_1,\boldsymbol{X}\rangle-b_1\\
\langle\boldsymbol{n}_1,\boldsymbol{\boldsymbol{n}_2}\rangle&\langle\boldsymbol{n}_2,\boldsymbol{X}\rangle-b_2
\end{vmatrix}}{\>\>\>\Vert\boldsymbol{n}_1\Vert^2\Vert\boldsymbol{n}_2\Vert^2-\langle\boldsymbol{n}_1,\boldsymbol{n}_2\rangle^2}\,\boldsymbol{n}_2\right\Vert=R
\end{align*}

1. (Det[( {
2.           {alpha1.u - b1, alpha1.alpha2},
3.           {alpha2.u - b2, alpha2.alpha2}
4.          } )]/((alpha1.alpha1)*(alpha2.alpha2) - (alpha1.alpha2)^2)*
5.        alpha1 + Det[( {
6.           {alpha1.alpha1, alpha1.u - b1},
7.           {alpha1.alpha2, alpha2.u - b2}
8.          } )]/((alpha1.alpha1)*(alpha2.alpha2) - (alpha1.alpha2)^2)*
9.        alpha2).(Det[( {
10.           {alpha1.u - b1, alpha1.alpha2},
11.           {alpha2.u - b2, alpha2.alpha2}
12.          } )]/((alpha1.alpha1)*(alpha2.alpha2) - (alpha1.alpha2)^2)*
13.        alpha1 + Det[( {
14.           {alpha1.alpha1, alpha1.u - b1},
15.           {alpha1.alpha2, alpha2.u - b2}
16.          } )]/((alpha1.alpha1)*(alpha2.alpha2) - (alpha1.alpha2)^2)*
17.        alpha2) - 9 /. {alpha1 -> {1, 2, -5}, alpha2 -> {2, 3, -6},
18.    b1 -> 3, b2 -> 7, u -> {x, y, z}} // Factor

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