# 7.5. 最小周长包围矩形

## 7.5. 最小周长包围矩形

$\left\{ {\begin{array}{*{20}{l}}{{A_A} = \sqrt 2 ab + 2{b^2}}\\{{P_A} = 2a + 4\sqrt 2 b}\\{{A_P} = \frac{{{a^2}}}{2} + \sqrt 2 ab + {b^2}}\\{{P_P} = 2\sqrt 2 a + 4b}\end{array}} \right.$

$\left\{ {\begin{array}{*{20}{l}}{{A_P} – {A_A} = \frac{{{a^2}}}{2} – {b^2} \succ 0}\\{{P_A} – {P_P} = 4(\sqrt 2 – 1)(b – \frac{a}{2}) \succ 0}\end{array}} \right.$

$\left\{ \begin{array}{l}{l_0} = {d_{i,k}} \cdot cos{\alpha _k}\\{l_1} = {d_{j,l}} \cdot cos{\alpha _l}\end{array} \right.,0 \le {\alpha _k} \prec \frac{\pi }{2}$

${D_{CC}} = 2({l_0}’ + {l_1}’)\left\{ {\begin{array}{*{20}{c}}{{l_0}’ = {d_{i,k}}\cos ({\alpha _k} – \delta )}\\{{l_1}’ = {d_{j,l}}\cos ({\alpha _l} + \delta )}\end{array}} \right.$

${D_C} = 2({l_0}” + {l_1}”)\left\{ {\begin{array}{*{20}{c}}{{l_0}” = {d_{i,k}}\cos ({\alpha _k} + \delta )}\\{{l_1}” = {d_{j,l}}\cos ({\alpha _l} – \delta )}\end{array}} \right.$

${D_{CC}} = 2({l_0} + {l_1})\cos \delta + 2({l_0}\tan {\alpha _k} – {l_1}\tan {\alpha _l})\sin \delta$

${D_{CC}} = D\cos \delta + C\sin \delta$

${D_C} = D\cos \delta – C\sin \delta$

$\begin{array}{l}\frac{{D\cos \delta – C\sin \delta }}{D} \ge 1\\ \Leftrightarrow \frac{C}{D}\sin \delta \le \cos \delta – 1\\ \Leftrightarrow \frac{{{D_{CC}}}}{D} \le 2\cos \delta – 1 \prec 1\end{array}$