点到双截面单纯形的投影

青青子衿

在$n$维欧式空间中，已知点$U$的坐标向量为$\,\boldsymbol{u}=(u_1,u_2,\cdots,u_n)$，给定双截面单纯形的第一截面法向量为$\,\boldsymbol{\alpha}_{1}=(a_{11},a_{12},\cdots,a_{1n})$，第一截面的截距常数为$b_1$，双截面单纯形的第二截面法向量为$\,\boldsymbol{\alpha}_{2}=(a_{21},a_{22},\cdots,a_{2n})$，第二截面的截距常数为$b_2$，双截面单纯形为
\begin{align*}
S&=\left\{\,\boldsymbol{x}|\,
\langle\boldsymbol{\alpha}_1,\boldsymbol{x}\rangle\leq\,\!b_1,\,\langle\boldsymbol{\alpha}_2,\boldsymbol{x}\rangle\leq\,\!b_2\\
\right\}\\
&=\left\{(x_1,x_2,\cdots,x_n)|\,
\substack{
a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\leq\,\!b_1\\
a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\leq\,\!b_2}\right\}\,;
\end{align*}
设点$U$在单截面上的投影坐标向量分别为
\begin{align*}
\boldsymbol{p}_1=\boldsymbol{u}-\dfrac{\langle\boldsymbol{\alpha}_1,\boldsymbol{u}\rangle-b_1}{\Vert\boldsymbol{\alpha}_1\Vert^2}\,\boldsymbol{\alpha}_1\\
\boldsymbol{p}_2=\boldsymbol{u}-\dfrac{\langle\boldsymbol{\alpha}_2,\boldsymbol{u}\rangle-b_2}{\Vert\boldsymbol{\alpha}_2\Vert^2}\,\boldsymbol{\alpha}_2\\
\end{align*}
设点$U$在双截面交线上的投影坐标向量为
\begin{align*}
\boldsymbol{p}_3=\boldsymbol{u}-\dfrac{\begin{vmatrix}
\langle\boldsymbol{\alpha}_1,\boldsymbol{u}\rangle-b_1&\langle\boldsymbol{\alpha}_1,\boldsymbol{\boldsymbol{\alpha}_2}\rangle\\
\langle\boldsymbol{\alpha}_2,\boldsymbol{u}\rangle-b_2&\Vert\boldsymbol{\alpha}_2\Vert^2
\end{vmatrix}}{\>\>\>\Vert\boldsymbol{\alpha}_1\Vert^2\Vert\boldsymbol{\alpha}_2\Vert^2-\langle\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2\rangle^2}\,\boldsymbol{\alpha}_1
-\dfrac{
\begin{vmatrix}
\Vert\boldsymbol{\alpha}_1\Vert^2&\langle\boldsymbol{\alpha}_1,\boldsymbol{u}\rangle-b_1\\
\langle\boldsymbol{\alpha}_1,\boldsymbol{\boldsymbol{\alpha}_2}\rangle&\langle\boldsymbol{\alpha}_2,\boldsymbol{u}\rangle-b_2
\end{vmatrix}}{\>\>\>\Vert\boldsymbol{\alpha}_1\Vert^2\Vert\boldsymbol{\alpha}_2\Vert^2-\langle\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2\rangle^2}\,\boldsymbol{\alpha}_2
\end{align*}
则点$u$到双截面单纯形的投影公式是一个分段函数
\begin{align*}
\operatorname{Proj}_S(\boldsymbol{u})=
\begin{cases}
\boldsymbol{u}&\,&\langle\boldsymbol{\alpha}_1,\boldsymbol{p}_2\rangle\leq\,\!b_1\wedge\langle\boldsymbol{\alpha}_2,\boldsymbol{p}_1\rangle\leq\,\!b_2\\
\boldsymbol{p}_1&\,&\langle\boldsymbol{\alpha}_1,\boldsymbol{p}_2\rangle\geq\,\!b_1\wedge\langle\boldsymbol{\alpha}_2,\boldsymbol{p}_1\rangle\leq\,\!b_2\\
\boldsymbol{p}_2&\,&\langle\boldsymbol{\alpha}_1,\boldsymbol{p}_2\rangle\leq\,\!b_1\wedge\langle\boldsymbol{\alpha}_2,\boldsymbol{p}_1\rangle\geq\,\!b_2\\
\boldsymbol{p}_3&\,&\langle\boldsymbol{\alpha}_1,\boldsymbol{p}_2\rangle\geq\,\!b_1\wedge\langle\boldsymbol{\alpha}_2,\boldsymbol{p}_1\rangle\geq\,\!b_2\\
\end{cases}
\end{align*}

\begin{align*}
p_{31}&=u_1-\dfrac{\begin{vmatrix}
a_{11}u_{1}+a_{12}u_{2}-b_{1}&
a_{11}a_{21}+a_{12}a_{22} \\
a_{21}u_{1}+a_{22}u_{2}-b_{2}& a_{21}^2+a_{22}^2 \\
\end{vmatrix}}{\begin{vmatrix}
a_{11}^2+a_{12}^2&
a_{11}a_{21}+a_{12}a_{22} \\
a_{11}a_{21}+a_{12}a_{22}& a_{21}^2+a_{22}^2 \\
\end{vmatrix}}a_{11}\\
&\qquad\qquad
-\dfrac{\begin{vmatrix}
a_{11}u_{1}+a_{12}u_{2}-b_{1}&
a_{11}a_{21}+a_{12}a_{22} \\
a_{21}u_{1}+a_{22}u_{2}-b_{2}& a_{21}^2+a_{22}^2 \\
\end{vmatrix}}{\begin{vmatrix}
a_{11}^2+a_{12}^2&
a_{11}a_{21}+a_{12}a_{22} \\
a_{11}a_{21}+a_{12}a_{22}& a_{21}^2+a_{22}^2 \\
\end{vmatrix}}a_{21}\\

p_{32}&=u_2-\dfrac{\begin{vmatrix}
a_{11}u_{1}+a_{12}u_{2}-b_{1}&
a_{11}a_{21}+a_{12}a_{22} \\
a_{21}u_{1}+a_{22}u_{2}-b_{2}& a_{21}^2+a_{22}^2 \\
\end{vmatrix}}{\begin{vmatrix}
a_{11}^2+a_{12}^2&
a_{11}a_{21}+a_{12}a_{22} \\
a_{11}a_{21}+a_{12}a_{22}& a_{21}^2+a_{22}^2 \\
\end{vmatrix}}a_{12}\\
&\qquad\qquad
-\dfrac{\begin{vmatrix}
a_{11}u_{1}+a_{12}u_{2}-b_{1}&
a_{11}a_{21}+a_{12}a_{22} \\
a_{21}u_{1}+a_{22}u_{2}-b_{2}& a_{21}^2+a_{22}^2 \\
\end{vmatrix}}{\begin{vmatrix}
a_{11}^2+a_{12}^2&
a_{11}a_{21}+a_{12}a_{22} \\
a_{11}a_{21}+a_{12}a_{22}& a_{21}^2+a_{22}^2 \\
\end{vmatrix}}a_{22}
\end{align*}

1. {u1 - Det[( {
2.        {a11*u1 + a12*u2 - b1, a11*a21 + a12*a22},
3.        {a21*u1 + a22*u2 - b2, a21^2 + a22^2}
4.       } )]/Det[( {
5.        {a11^2 + a12^2, a11*a21 + a12*a22},
6.        {a11*a21 + a12*a22, a21^2 + a22^2}
7.       } )] a11 - Det[( {
8.        {a11^2 + a12^2, a11*u1 + a12*u2 - b1},
9.        {a11*a21 + a12*a22, a21*u1 + a22*u2 - b2}
10.       } )]/Det[( {
11.        {a11^2 + a12^2, a11*a21 + a12*a22},
12.        {a11*a21 + a12*a22, a21^2 + a22^2}
13.       } )] a21,
14.   u2 - Det[( {
15.        {a11*u1 + a12*u2 - b1, a11*a21 + a12*a22},
16.        {a21*u1 + a22*u2 - b2, a21^2 + a22^2}
17.       } )]/Det[( {
18.        {a11^2 + a12^2, a11*a21 + a12*a22},
19.        {a11*a21 + a12*a22, a21^2 + a22^2}
20.       } )] a12 - Det[( {
21.        {a11^2 + a12^2, a11*u1 + a12*u2 - b1},
22.        {a11*a21 + a12*a22, a21*u1 + a22*u2 - b2}
23.       } )]/Det[( {
24.        {a11^2 + a12^2, a11*a21 + a12*a22},
25.        {a11*a21 + a12*a22, a21^2 + a22^2}
26.       } )] a22} /. {u1 -> 4, u2 -> 4,
27.   a11 -> 3, a12 -> 2, b1 -> 7, a21 -> 5, a22 -> 9, b2 -> 18}

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