# 二面角

## 立体几何

${\displaystyle \because AB\perp OB}$

${\displaystyle \therefore OB}$ ${\displaystyle OA}$ 在平面 ${\displaystyle \beta }$  中的射影

${\displaystyle \because OA\perp l,OB\perp l}$

${\displaystyle \therefore \angle AOB}$  是二面角${\displaystyle \alpha -l-\beta }$  的平面角。

## 解析几何

${\displaystyle a_{1}x+b_{1}y+c_{1}z+d_{1}=0}$
${\displaystyle a_{2}x+b_{2}y+c_{2}z+d_{2}=0\,\,,}$

${\displaystyle \cos \varphi ={\frac {\left\vert a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}\right\vert }{{\sqrt {a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}{\sqrt {a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}}\,\,.}$

${\displaystyle \cos \varphi ={\frac {\left\vert \mathbf {n} _{\mathrm {A} }\cdot \mathbf {n} _{\mathrm {B} }\right\vert }{|\mathbf {n} _{\mathrm {A} }||\mathbf {n} _{\mathrm {B} }|}}\,\,,}$

${\displaystyle \varphi =\operatorname {atan2} \left({\big (}[\mathbf {b} _{1}\times \mathbf {b} _{2}]\times [\mathbf {b} _{2}\times \mathbf {b} _{3}]{\big )}\cdot {\frac {\mathbf {b} _{2}}{|\mathbf {b} _{2}|}}\,\,,\,\,[\mathbf {b} _{1}\times \mathbf {b} _{2}]\cdot [\mathbf {b} _{2}\times \mathbf {b} _{3}]\right)\,\,,}$

## 立体化学

 构象名称 syn-正丁烷纽曼投影 syn-正丁烷锯木架投影

## 几何

${\displaystyle \cos \varphi ={\frac {\cos(\angle \mathrm {APB} )-\cos(\angle \mathrm {APC} )\cos(\angle \mathrm {BPC} )}{\sin(\angle \mathrm {APC} )\sin(\angle \mathrm {BPC} )}}\,\,.}$

## 参考资料

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