広告
・3角形要素の面積
3角形要素の面積は次式となります。
\[
S=\frac{1}{2}
\begin{vmatrix}
1&x_1&y_1\\
1&x_2&y_2\\
1&x_3&y_3
\end{vmatrix}
\]
行列式は下記で計算されます。
\[
\det A=\sum_{\sigma\in\sigma_n}\varepsilon(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)}
\]
\(\varepsilon(\sigma)\)を \(\sigma\in\sigma_n\) の符号とすると
(1) \(\varepsilon(\tau\sigma)=\varepsilon(\sigma)\varepsilon(\tau)\)
(2) \(\varepsilon(1)=1\)
(3) \(\varepsilon(\sigma^{-1})=\varepsilon(\sigma)\)
(4) \(\rho\) が互換であれば \(\varepsilon(\rho)=-1\)
\[
\begin{aligned}
\det
\begin{pmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}
\end{pmatrix}
&=
\sum_{\sigma\in\sigma_3}\varepsilon(\sigma)
a_{1\sigma(1)}a_{2\sigma(2)}a_{3\sigma(3)}
\\[6pt]
&=
\sum_{\sigma\in\sigma_3}
\varepsilon\!\left(
\begin{pmatrix}1&2&3\\\sigma(1)&\sigma(2)&\sigma(3)\end{pmatrix}
\right)
a_{1\sigma(1)}a_{2\sigma(2)}a_{3\sigma(3)}
\\[6pt]
&=
\varepsilon\!\left(\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\right)a_{11}a_{22}a_{33}
+
\varepsilon\!\left(\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}\right)a_{11}a_{23}a_{32}
\\[6pt]
&\quad+
\varepsilon\!\left(\begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}\right)a_{12}a_{21}a_{33}
+
\varepsilon\!\left(\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}\right)a_{12}a_{23}a_{31}
\\[6pt]
&\quad+
\varepsilon\!\left(\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}\right)a_{13}a_{21}a_{32}
+
\varepsilon\!\left(\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}\right)a_{13}a_{22}a_{31}
\\[6pt]
&=
\varepsilon\!\left(\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\right)a_{11}a_{22}a_{33}
+
\varepsilon\!\left((2\ 3)\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\right)a_{11}a_{23}a_{32}
\\[6pt]
&\quad+
\varepsilon\!\left((1\ 2)\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\right)a_{12}a_{21}a_{33}
+
\varepsilon\!\left((1\ 2)(2\ 3)\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\right)a_{12}a_{23}a_{31}
\\[6pt]
&\quad+
\varepsilon\!\left((1\ 3)(2\ 3)\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\right)a_{13}a_{21}a_{32}
+
\varepsilon\!\left((1\ 3)\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}\right)a_{13}a_{22}a_{31}
\\[6pt]
&=
(-1)^0a_{11}a_{22}a_{33}
+(-1)^1a_{11}a_{23}a_{32}
\\[6pt]
&\quad+
(-1)^1a_{12}a_{21}a_{33}
+(-1)^2a_{12}a_{23}a_{31}
\\[6pt]
&\quad+
(-1)^2a_{13}a_{21}a_{32}
+(-1)^1a_{13}a_{22}a_{31}
\\[6pt]
&=
a_{11}a_{22}a_{33}
-a_{11}a_{23}a_{32}
\\[6pt]
&\quad-
a_{12}a_{21}a_{33}
+a_{12}a_{23}a_{31}
\\[6pt]
&\quad+
a_{13}a_{21}a_{32}
-a_{13}a_{22}a_{31}
\end{aligned}
\]
従って、3角形の面積Sは、
\[
\begin{aligned}
S&=\frac{1}{2}\det
\begin{pmatrix}
1&x_1&y_1\\
1&x_2&y_2\\
1&x_3&y_3
\end{pmatrix}\\[6pt]
&=\frac{1}{2}\left(
x_2y_3-y_2x_3-x_1y_3+x_1y_2+y_1x_3-y_1x_2
\right)
\end{aligned}
\]
\[
\begin{aligned}
&a_{11}=1,\ a_{12}=x_1,\ a_{13}=y_1\\[6pt]
&a_{21}=1,\ a_{22}=x_2,\ a_{23}=y_2\\[6pt]
&a_{31}=1,\ a_{32}=x_3,\ a_{33}=y_3
\end{aligned}
\]
| prev | | | up | | | next |
