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mathoverflow.net/questions/336464/a-curious-r … ges-of-a-tetrahedron
考虑一个四面体，其棱长为
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
二面角为
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \alpha_{34}.
$$
立体角为
\begin{split}
&\Omega_1=\alpha_{12}+\alpha_{13}+\alpha_{14}-\pi \\
&\Omega_2=\alpha_{12}+\alpha_{23}+\alpha_{24}-\pi \\
&\Omega_3=\alpha_{13}+\alpha_{23}+\alpha_{34}-\pi \\
&\Omega_4=\alpha_{14}+\alpha_{24}+\alpha_{34}-\pi \\
\end{split}各面的周长为
\begin{split}
&P_1=l_{23}+l_{34}+l_{24} \\
&P_2=l_{13}+l_{14}+l_{34} \\
&P_3=l_{12}+l_{14}+l_{24} \\
&P_4=l_{12}+l_{23}+l_{13}. \\
\end{split}那么以下交比相等：
$$
[e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[P_1, P_2, P_3, P_4].
$$
其中，“交比”$[a,b,c,d]$的定义为$[a,b,c,d]=\frac{(a-c)(b-d)}{(a-d)(b-c)}$. |
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