広告
・3角形要素の面積
3角形要素の面積を示します。
\[
\begin{aligned}
S&=\frac{1}{2}\det
\begin{pmatrix}
1&x_1&y_1\\
1&x_2&y_2\\
1&x_3&y_3
\end{pmatrix}\\[6pt]
&=\frac{1}{2}
(x_2y_3-y_2x_3-x_1y_3+x_1y_2+y_1x_3-y_1x_2)
\end{aligned}
\]
ここで、
\[
\begin{aligned}
a_{11}&=1,\ a_{12}=x_1,\ a_{13}=y_1\\[6pt]
a_{21}&=1,\ a_{22}=x_2,\ a_{23}=y_2\\[6pt]
a_{31}&=1,\ a_{32}=x_3,\ a_{33}=y_3
\end{aligned}
\]
同様に
\[
\begin{aligned}
S_1&=\frac{1}{2}\det
\begin{pmatrix}
1&x&y\\
1&x_2&y_2\\
1&x_3&y_3
\end{pmatrix}\\[6pt]
&=\frac{1}{2}
(x_2y_3-y_2x_3-xy_3+xy_2+yx_3-yx_2)
\end{aligned}
\]
\[
\begin{aligned}
S_2&=\frac{1}{2}\det
\begin{pmatrix}
1&x_1&y_1\\
1&x&y\\
1&x_3&y_3
\end{pmatrix}\\[6pt]
&=\frac{1}{2}
(xy_3-yx_3-x_1y_3+x_1y+y_1x_3-y_1x)
\end{aligned}
\]
\[
\begin{aligned}
S_3&=\frac{1}{2}\det
\begin{pmatrix}
1&x_1&y_1\\
1&x_2&y_2\\
1&x&y
\end{pmatrix}\\[6pt]
&=\frac{1}{2}
(x_2y-y_2x-x_1y+x_1y_2+y_1x-y_1x_2)
\end{aligned}
\]
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