凸体的包含测度问题是积分几何中相当重要的课题之一.任德麟在80年代建立了二维和n维含于凸体内定长线段的的运动测度的系统理论, 推导出n维欧式空间中凸体的弦幂积分不等式, 提出了并解决了一系列复杂的几何概率课题^[1-2].Santalo将平行线网格推广到平行带域网格, 同时将小针推广到凸域, 但他只研究了凸域直径不超过带域间距离的情况.任德麟作出了进一步推广, 取消凸域直径不超过带域的间距离这一限制, 而后其学生黎荣泽、张高勇讨论了相交的两组平行线网上的Buffon概率, 本文拟研究以正六边形和菱形为基本区域的复合网格中的Buffon问题.

定义2.1 以$\sigma$表示凸域$D$被直线$G$截出的弦长.当$G$仅与$\partial$$D$相交包括$G$是线段情形, 约定$\sigma$=0, $G$的表示取广义法式, 对任意给定的$\sigma$和${\varphi(0 \le \varphi \le 2\pi )}$, 令

$\begin{equation} {p\left(\sigma, \varphi \right)}= \mathop {\sup }\limits_G {\{ p:m[G \cap \left({\mathop{\rm int}} D\right)] = \sigma} \}, \end{equation}$

(2.1)

称二元函数${p\left(\sigma, \varphi \right)}$为凸域的广义支持函数^[1-3].

定义2.2 以${\sigma _M}(\varphi )$表示垂直于$\varphi$方向的直线$G$与凸域$D$截出的弦长最大值, 即

$\begin{equation} {\sigma _M}(\varphi ) = \mathop {\sup }\limits_G \{ \sigma :\sigma = m[G \cap ({\mathop{\rm int}} D)]\}. \end{equation}$

(2.2)

对任意给定的$l(l \ge 0)$及${\varphi(0 \le \varphi \le 2\pi )}$, 令

$\begin{equation} {r(l, \varphi )} = \min \{ l, {\sigma _M}(\varphi )\}, \end{equation}$

(2.3)

称二元函数$r(l, \varphi )$为凸域$D$的限弦函数^[1-3].

考虑将长为$l$的小针投掷于以正六边形${K_1}$和菱形${K_2}$为基本区域的平面网格(如图 1所示)中, 研究小针与此网格相交的概率.

设${K_1}$、${K_2}$的面积和周长分别为${F_1}$、${F_2}$和${L_1}$、${L_2}$, 含于${K_1}$内小针的运动测度为${m_1}(l)$, 含于${K_2}$内小针的运动测度为$ {m_2}(l)$.

假设正六边形的边长为$a$, 则${F_1} = \frac{{3\sqrt 3 }}{2}{a^2}$, ${F_2} = \frac{{\sqrt 3 }}{2}{a^2}$.

取${n^2}$个边长为$a$的正方形组成以$na$为边长的大正方形, 含于大正方形内小针的运动测度为记为${m_3}(l)$, 当$n \to \infty $时

$\begin{equation} p = \mathop {\lim }\limits_{n \to \infty } \frac{{{m_3}(l) - \frac{{{{(na)}^2}}}{{{F_1} + {F_2}}}({m_1}(l) + {m_2}(l))}}{{{m_3}(l)}}, \end{equation}$

(3.1)

即为小针与此网格相遇的概率^[5].

引理3.1 设${p\left(\sigma, \varphi \right)}$和$r(l, \varphi )$分别为凸域$D$的广义支撑函数和限弦函数, ${m}(l)$之定义同前, 则有

$\begin{equation} m(l) = \pi F - \int_0^{2\pi } {d\varphi } \int_0^{r(l, \varphi )} {p(\sigma, \varphi )} d\sigma, \end{equation}$

(3.2)

其中$F$为$D$之面积^[1].

对于边长为$a$一个角为$\pi /3$的菱形建立如下坐标系

则此菱形的广义支撑函数为

$ {p_1} = \frac{{\sqrt 3 }}{2}a\cos \varphi - \frac{{\sqrt 3 }}{6}\sigma (3{\cos ^2}\varphi - {\sin ^2}\varphi ), \varphi \in\left[{0, \frac{\pi }{3}} \right), \\ {p_2} = \frac{1}{2}a\sin \varphi + \frac{{\sqrt 3 }}{6}\sigma (3{\cos ^2}\varphi- {\sin ^2}\varphi ), \varphi \in\left[{\frac{\pi }{3}, \frac{\pi }{2}} \right]. $

限弦函数为

$r(l, \varphi ) = \left\{ \begin{array}{l} l, \quad \quad \quad \quad \quad \quad \quad \quad \quad l \in [0, \frac{{\sqrt 3 a}}{2}), \varphi \in [0, \frac{\pi }{2}], \\ l, \quad \quad \quad \quad \quad \quad \quad \quad \quad l \in [\frac{{\sqrt 3 a}}{2}, a), \varphi \in [0, \frac{\pi }{6}-\arccos \frac{{\sqrt 3 a}}{{2l}}], \\ \frac{{\sqrt 3 a}}{{2\cos (\varphi - \pi /6) {\kern 1pt} {\kern 1pt} {\kern 1pt} }}, \quad \quad \quad \quad \;l \in [\frac{{\sqrt 3 a}}{2}, a), \varphi \in (\frac{\pi }{6}-\arccos \frac{{\sqrt 3 a}}{{2l}}, \frac{\pi }{6} + \arccos \frac{{\sqrt 3 a}}{{2l}}], \\ l, \quad \quad \quad \quad \quad \quad \quad \quad \quad l \in [\frac{{\sqrt 3 a}}{2}, a), \varphi \in (\frac{\pi }{6} + \arccos \frac{{\sqrt 3 a}}{{2l}}, \frac{\pi }{3}]{\kern 1pt} {\kern 1pt} {\kern 1pt}, \\ l, \quad \quad \quad \quad \quad \quad \quad \quad \quad l \in [\frac{{\sqrt 3 a}}{2}, a), \varphi \in (\frac{\pi }{3}, \frac{\pi }{2}]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}, \\ \frac{{\sqrt 3 a}}{{2\cos (\varphi - \pi /6) {\kern 1pt} {\kern 1pt} {\kern 1pt} }}, \quad \quad \quad \quad {\kern 1pt} \;l \in [a, \sqrt 3 a], \varphi \in (\frac{\pi }{3}, \frac{\pi }{6} + \arccos \frac{{\sqrt 3 a}}{{2l}}], {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ l, \quad \quad \quad \quad \quad \quad \quad \quad \quad l \in [a, \sqrt 3 a], \varphi \in (\frac{\pi }{6} + \arccos \frac{{\sqrt 3 a}}{{2l}}, \frac{\pi }{2}]. {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \end{array} \right.$

设$I = \int_0^{\frac{\pi }{2}} {d\varphi \int_0^{r(l, \varphi )} {pd\sigma } } $, 则分三种情况讨论

1. ${\kern 1pt} {\kern 1pt} l \in [0, \frac{{\sqrt 3 a}}{2})$,

${I_1} = \int_0^{\frac{\pi }{2}} {d\varphi \int_0^l {pd\sigma } } = \int_0^{\frac{\pi }{3}} {d\varphi } \int_0^l {{p_1}d\sigma } + \int_{\frac{\pi }{3}}^{\frac{\pi }{2}} {d\varphi } \int_0^l {{p_2}d\sigma } = al- \frac{{{l^2}}}{4}- \frac{{\sqrt 3 }}{{72}}\pi {l^2}.$

2. ${\kern 1pt} {\kern 1pt} l \in [\frac{{\sqrt 3 a}}{2}, a)$,

$ {I_2} = \int_0^{\frac{\pi }{6}- \arccos \frac{{\sqrt 3 a}}{{2l}}} {d\varphi } \int_0^l {{p_1}d\sigma } + \int_{\frac{\pi }{6}- \arccos \frac{{\sqrt 3 a}}{{2l}}}^{\frac{\pi }{6} + \arccos \frac{{\sqrt 3 a}}{{2l}}} {d\varphi } \int_0^{\frac{{\sqrt 3 a}}{{2\cos \varphi }}} {{p_1}d\sigma } + \int_{\frac{\pi }{6} + \arccos \frac{{\sqrt 3 a}}{{2l}}}^{\frac{\pi }{3}} {d\varphi } \int_0^l {{p_1}d\sigma } \\ + \int_{\frac{\pi }{3}}^{\frac{\pi }{2}} {d\varphi } \int_0^l {{p_2}d\sigma } \\ = al- \frac{{{l^2}}}{4} - \frac{{\sqrt 3 }}{{72}}\pi {l^2} - \frac{5}{4}a\sqrt {{l^{^2}} - \frac{3}{4}{a^2}} + (\frac{{\sqrt 3 }}{6}{l^2} + \frac{{\sqrt 3 }}{2}{a^2})\arccos \frac{{\sqrt 3 a}}{{2l}}. $

3. ${\kern 1pt} l \in [a, \sqrt 3 a]$,

$ {I_3} = \int_0^{\frac{\pi }{3}} {d\varphi } \int_0^{\frac{h}{{\cos (\varphi - \pi /6)}}} {{p_1}d\sigma } + \int_{\frac{\pi }{3}}^{\frac{\pi }{6} + \arccos \frac{{\sqrt 3 a}}{{2l}}} {d\varphi } \int_a^{\frac{{\sqrt 3 a}}{{2\cos (\varphi - \pi /6) {\kern 1pt} {\kern 1pt} {\kern 1pt} }}{\kern 1pt} } {{p_2}d\sigma } + \int_{\frac{\pi }{6} + \arccos \frac{{\sqrt 3 a}}{{2l}}}^{\frac{\pi }{2}} {d\varphi } \int_a^l {{p_2}d\sigma } \\ = \frac{{\sqrt 3 }}{{24}}\pi {a^2} + \frac{{\sqrt 3 }}{{36}}\pi {l^2} + \frac{3}{{16}}{a^2} + \frac{{{l^2}}}{8} - \frac{3}{8}a\sqrt {{l^{^2}} - \frac{3}{4}{a^2}} + (\frac{{\sqrt 3 }}{4}{a^2} - \frac{{\sqrt 3 }}{{12}}{l^2})\arccos \frac{{\sqrt 3 a}}{{2l}}. $

于是此菱形的运动测度为

${m_2}(l) = \frac{{\sqrt 3 }}{2}\pi {a^2} - 4I, $

将前面的结果代入可得

${m_2}(l) = \left\{ \begin{array}{l} \frac{{\sqrt 3 }}{2}\pi {a^2} - 4al + {l^2} + \frac{{\sqrt 3 }}{{18}}\pi {l^2}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l \in [0, \frac{{\sqrt 3 }}{2}a), \\ \frac{{\sqrt 3 }}{2}\pi {a^2}- 4al + {l^2} + \frac{{\sqrt 3 }}{{18}}\pi {l^2} + 5a\sqrt {{l^{^2}}- \frac{3}{4}{a^2}}, \\- (\frac{{2\sqrt 3 }}{3}{l^2} + 2\sqrt 3 {a^2})\arccos \frac{{\sqrt 3 a}}{{2l}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l \in [\frac{{\sqrt 3 }}{2}a, a){\kern 1pt} {\kern 1pt} {\kern 1pt}, \\ \frac{{\sqrt 3 }}{3}\pi {a^2}- \frac{3}{4}{a^2}- \frac{1}{2}{l^2}- \frac{{\sqrt 3 }}{9}\pi {l^2}{\rm{ + }}\frac{3}{2}a\sqrt {{l^{^2}} - \frac{3}{4}{a^2}}, \\ - (\sqrt 3 {a^2} - \frac{{\sqrt 3 }}{3}{l^2})\arccos \frac{{\sqrt 3 a}}{{2l}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l \in [a, \sqrt 3 a]{\kern 1pt} {\kern 1pt}. \end{array} \right.$

由上述方法还可以得到

正六边形内定长线段的运动测度^[4]

${m_1}(l) = \left\{ \begin{array}{l} \frac{{3\sqrt 3 }}{2}\pi {a^2} - 6al - \frac{{\sqrt 3 \pi {l^2}}}{6} + \frac{3}{2}{l^2}, \quad \quad \quad \quad \quad \quad \quad \;\quad \;l \in [0, a], {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ \frac{{5\sqrt 3 }}{2}\pi {a^2} + \frac{{\sqrt 3 \pi {l^2}}}{2} -(3\sqrt 3 {a^2} + 2\sqrt 3 {l^2}), \\ \arcsin \frac{{\sqrt 3 a}}{{2l}} -\frac{{9a}}{2}\sqrt {4{l^2} -3{a^2}}, \quad \quad \quad \quad \quad \quad \quad \;\quad \;\;l \in (a, \sqrt 3 a], {\kern 1pt} \\ 2\sqrt 3 \pi {a^2} + \frac{{\sqrt 3 }}{6}\pi {l^2} -9{a^2} -\frac{3}{2}{l^2} + 15a\sqrt {{l^2} -3{a^2}}, \\ - (12\sqrt 3 {a^2} + \sqrt 3 {l^2})\arccos \frac{{\sqrt 3 a}}{l}, \quad \quad \quad \quad \quad \quad \;\, \quad l \in (\sqrt 3 a, 2a].{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \end{array} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} $

大正方形内定长线段的运动测度

${m_3}\left( l \right) = \pi {n^2}{a^2} - 4nal + {l^2}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l \in [0, na), $

则小针与网格相交的概率

$p = \mathop {\lim }\limits_{n \to \infty } \frac{{{m_3}(l)- \frac{{{{(na)}^2}}}{{{F_1} + {F_2}}}({m_1}(l) + {m_2}(l))}}{{{m_3}(l)}} = 1- \frac{{({m_1}(l) + {m_2}(l))}}{{\pi ({F_1} + {F_2})}} = 1- \frac{{({m_1}(l) + {m_2}(l))}}{{2\sqrt 3 \pi {a^2}}}.$

(i)当$l \in [0, \frac{{\sqrt 3 }}{2}a)$时,

$p = \frac{{10al- \frac{5}{2}{l^2} + \frac{{\sqrt 3 }}{9}\pi {l^2}}}{{2\sqrt 3 \pi {a^2}}}. $

(ii)当$l \in [\frac{{\sqrt 3 }}{2}a, a)$时,

$p = \frac{{10al- \frac{5}{2}{l^2} + \frac{{\sqrt 3 }}{9}\pi {l^2}- \frac{5}{4}a\sqrt {{l^{^2}}- \frac{3}{4}{a^2}} + (\frac{{\sqrt 3 }}{6}{l^2} + \frac{{\sqrt 3 }}{2}{a^2})\arccos \frac{{\sqrt 3 a}}{{2l}}}}{{2\sqrt 3 \pi {a^2}}}. $

(iii)当$l \in [a, \sqrt 3 a]$时,

$p =\frac{\frac{3}{4}{a^2} + \frac{1}{2}{l^2} - \frac{{5\sqrt 3 }}{6}\pi {a^2} - \frac{{7\sqrt 3 }}{{18}}\pi {l^2} + \frac{{15}}{2}a\sqrt {{l^{^2}} - \frac{3}{4}{a^2}} + (\sqrt 3 {a^2} - \frac{{\sqrt 3 }}{3}{l^2})\arccos \frac{{\sqrt 3 a}}{{2l}}}{{{2\sqrt 3 \pi {a^2}}}} \\ +\frac{ (3\sqrt 3 {a^2} + 2\sqrt 3 {l^2})\arcsin \frac{{\sqrt 3 a}}{{2l}}}{ {{2\sqrt 3 \pi {a^2}}}}.$

(iv)当$l \in [\sqrt 3 a, 2a]$时,

$p = \frac{{9{a^2} + \frac{3}{2}{l^2} - \frac{{\sqrt 3 }}{6}\pi {l^2} - 15a\sqrt {{l^2} - 3{a^2}} + (12\sqrt 3 {a^2} + \sqrt 3 {l^2})\arccos \frac{{\sqrt 3 a}}{l}}}{{2\sqrt 3 \pi {a^2}}}. $