# 6.4. 多边形的面积

## 6.4. 多边形的面积

#### 6.4.1. 二维空间

${\mathop{\rm Area}\nolimits} ({\rm P}) = \frac{1}{2}\sum\limits_{i = 0}^{n – 1} {{x_i}({y_{i + 1}} – {y_{i – 1}})} \tag{6.3}$

${\mathop{\rm Area}\nolimits} ({\rm P}) = {\mathop{\rm Area}\nolimits} (O{V_0}{V_1}) + {\mathop{\rm Area}\nolimits} (O{V_1}{V_2}) + \cdots + {\mathop{\rm Area}\nolimits} (O{V_{n – 1}}{V_0}) \tag{6.4}$

$Area(R) = \frac{1}{2}\sum\limits_{i = 0}^{n – 1} {\left( {{x_i}{y_{i + 1}} – {x_{i + 1}}{y_i}} \right)} \tag{6.5}$

$\begin{array}{l}{\rm{ }}\frac{1}{2}\sum\limits_{i = 0}^{n – 1} {\left( {{x_i}{y_{i + 1}} – {x_{i + 1}}{y_i}} \right)} \\ = \frac{1}{2}\sum\limits_{i = 0}^{n – 1} {\left[ {{x_i}\left( {{y_{i + 1}} – {y_{i – 1}}} \right) + {x_i}{y_{i – 1}} – {x_{i + 1}}{y_i}} \right]} \\ = \frac{1}{2}\sum\limits_{i = 0}^{n – 1} {{x_i}\left( {{y_{i + 1}} – {y_{i – 1}}} \right)} + \sum\limits_{i = 0}^{n – 1} {\left( {{x_i}{y_{i – 1}} – {x_{i + 1}}{y_i}} \right)} \\ = \frac{1}{2}\sum\limits_{i = 0}^{n – 1} {{x_i}\left( {{y_{i + 1}} – {y_{i – 1}}} \right)} + \left( {{x_0}{y_{ – 1}} – {x_n}{y_{n – 1}}} \right)\\ = \frac{1}{2}\sum\limits_{i = 0}^{n – 1} {{x_i}\left( {{y_{i + 1}} – {y_{i – 1}}} \right)} \end{array}$

#### 6.4.2. 三维空间

Goldman(1991)[23]给出了一个面积的计算公式：

${\rm{Area}}(R) = \frac{1}{2}\vec n \cdot \sum\limits_{i = 0}^{n – 1} {\left( {{V_i} \times {V_{i + 1}}} \right)} \tag{6.6}$

${\mathop{\rm Area}\nolimits} (\Delta O'{V_i}{V_{i + 1}}) = \frac{1}{2}\left\| {\overline {O’Q} } \right\| \cdot \left\| {\overline {{V_i}{V_{i + 1}}} } \right\| = \frac{1}{2}\left\| {\overline {OQ} } \right\| \cdot \left\| {\overline {{V_i}{V_{i + 1}}} } \right\| \cdot \cos \theta = \vec n \cdot \vec m \tag{6.7}$