Snell’s Law
$$
\begin{align}
\eta{_1} \sin(\theta_1) &\quad = \eta{_2}\sin(\theta{_2}) & n &\quad = (0,1) \notag \\[2ex]
L\qquad &\quad = (-\sin(\theta_1) , \cos(\theta_1)) & T &\quad = (\sin(\theta_2) , -\cos(\theta_2))\notag
\end{align}
$$
设:
$$\overrightarrow{T} = x \times \overrightarrow{L} + y \times \overrightarrow{n}$$
$$
\begin{align}
\Rightarrow
& \begin{cases}
&\sin(\theta_2) &= &\quad -\sin(\theta_1) \times x + 0 \times y \notag \\[2ex]
&-\cos(\theta_2) &= &\quad \cos(\theta_1) \times x + 1 \times y \notag
\end{cases} \\[2ex]
\Rightarrow
& \begin{cases}
x &= -\frac{\sin(\theta_2)}{\sin(\theta_1)} \notag \\[2ex]
y &= \cos(\theta_1) * \frac{\sin(\theta_2)}{\sin(\theta_1)} - \cos(\theta_2) \notag
\end{cases}
\end{align}
$$
$$
\begin{align}
\Rightarrow
\overrightarrow{T} &= -\frac{\sin(\theta_2)}{\sin(\theta_1)} \times \overrightarrow{L} + (\cos(\theta_1) \times \frac{\sin(\theta_2)}{\sin(\theta_1)} - \cos(\theta_2)) \times \overrightarrow{n} \notag \\[2ex]
\because \cos(\theta_2) &= \sqrt{1 - \sin^2(\theta_2)} \notag \\
&= \sqrt{1 - \frac{\sin^2(\theta_2)}{\sin^2(\theta_1)}(1 - \cos^2(\theta_1))} \notag \\
&= \sqrt{1 - \frac{\sin^2(\theta_2)}{\sin^2(\theta_1)}(1 - (\overrightarrow{L} \cdot \overrightarrow{n})^2)}\notag
\end{align}
$$
$$
\begin{align}
\Rightarrow \overrightarrow{T} &= -\frac{\sin(\theta_2)}{\sin(\theta_1)} \times \overrightarrow{L} + (\cos(\theta_1) \times \frac{\sin(\theta_2)}{\sin(\theta_1)} - \sqrt{1 - \frac{\sin^2(\theta_2)}{\sin^2(\theta_1)}(1 - (\overrightarrow{L} \cdot \overrightarrow{n})^2)}) \times \overrightarrow{n}\notag \\[2ex]
&= -\frac{\eta_1}{\eta_2} \times \overrightarrow{L} + \overrightarrow{n}(\frac{\eta_1}{\eta_2}(\overrightarrow{L} \cdot \overrightarrow{n}) - \sqrt{1 - \frac{\eta_1^2}{\eta_2^2}(1 - (\overrightarrow{L} \cdot \overrightarrow{n})^2)})\notag
\end{align}
$$
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