光线折射公式推导：Snell‘s Law

光线折射公式推导：Snell’s Law

设 入射光 $\mathbf{i}$ 和 折射光 $\mathbf{t}$ 均为单位向量，由Snell’s law可知 ${\eta }_i\sin {\theta }_i={\eta }_t\sin {\theta }_t$ 。

其中 ${\eta }_i,{\eta }_t$ 为反射系数（typically air = 1.0, glass = 1.3–1.7, diamond = 2.4）。

因为 $\sin {\theta }_i=\frac{|{\mathbf{i}}_{\parallel }|}{|\mathbf{i}|},\sin {\theta }_t=\frac{|{\mathbf{t}}_{\parallel }|}{|\mathbf{t}|}$ ，故：
η i ∣ i ∥ ∣ ∣ i ∣ = η t ∣ t ∥ ∣ ∣ t ∣ η i ∣ i ∥ ∣ = η t ∣ t ∥ ∣

ηi​∣i∣∣i∥​∣​=ηt​∣t∣∣t∥​∣​ηi​∣i∥​∣=ηt​∣t∥​∣​
又因 $\mathbf{i}$ 和 $\mathbf{t}$ 平行且方向相同：
η i ⋅ i ∥ = η t ⋅ t ∥ t ∥ = η i η t ⋅ i ∥ = η i η t ( i + cos ⁡ θ i ⋅ n )

ηi⋅i∥=ηt⋅t∥t∥=ηiηt⋅i∥=ηiηt(i+cos⁡θi⋅n)" role="presentation" style="position: relative;">ηi⋅i∥t∥===ηt⋅t∥ηiηt⋅i∥ηiηt(i+cosθi⋅n)$$\begin{array}{cccc}{\eta }_i\cdot {\mathbf{i}}_{\parallel }& =& {\eta }_t\cdot {\mathbf{t}}_{\parallel }& \\ {\mathbf{t}}_{\parallel }& =& \frac{{\eta }_i}{{\eta }_t}\cdot {\mathbf{i}}_{\parallel }\\ & =& \frac{{\eta }_i}{{\eta }_t}(\mathbf{i}+\cos {\theta }_i\cdot \mathbf{n})\end{array}$$

ηi​⋅i∥​t∥​​===​ηt​⋅t∥​ηt​ηi​​⋅i∥​ηt​ηi​​(i+cosθi​⋅n)​
其中 ${\mathbf{i}}_{\parallel }=\frac{\mathbf{i}\cdot \mathbf{n}}{|\mathbf{i}||\mathbf{n}|}\mathbf{n}=\cos {\theta }_i\cdot \mathbf{n}$.

又因 $\mathbf{t}={\mathbf{t}}_{\perp }+{\mathbf{t}}_{\parallel }$ 且 $|\mathbf{t}{|}^2=|{\mathbf{t}}_{\perp }{|}^2+|{\mathbf{t}}_{\parallel }{|}^2$ ：
∣ t ⊥ ∣ 2 = ∣ t ∣ 2 − ∣ t ∥ ∣ 2 ∣ t ⊥ ∣ = 1 − ∣ t ∥ ∣ 2 t ⊥ = − 1 − ∣ t ∥ ∣ 2 ⋅ n

∣t⊥​∣2∣t⊥​∣t⊥​​===​∣t∣2−∣t∥​∣21−∣t∥​∣2 ​−1−∣t∥​∣2 ​⋅n​
从而：
$$\mathbf{t}={\mathbf{t}}_{\parallel }+{\mathbf{t}}_{\perp }$$

// r_in 为射入物体的光线
vec3 refract(const vec3& r_in, const vec3& outward_normal, double etai_over_etat) {
    double cos_thetai = dot(-r_in, outward_normal);
    vec3 r_out_parallel =  etai_over_etat * (r_in + cos_thetai * outward_normal);
    vec3 r_out_perp = -sqrt(1.0 - r_out_parallel.length_squared()) * outward_normal;
    return r_out_perp + r_out_parallel;
}
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也可进一步化简：
t = t ∥ + t ⊥ = η i η t ⋅ ( i + cos ⁡ θ i ⋅ n ) − 1 − ∣ i ∥ ∣ 2 ⋅ n = η i η t ⋅ ( i + cos ⁡ θ i ⋅ n ) − 1 − sin ⁡ 2 θ t ⋅ n = η i η t ⋅ i + ( η i η t cos ⁡ θ i − 1 − sin ⁡ 2 θ t ) ⋅ n

t​====​t∥​+t⊥​ηt​ηi​​⋅(i+cosθi​⋅n)−1−∣i∥​∣2 ​⋅nηt​ηi​​⋅(i+cosθi​⋅n)−1−sin2θt​ ​⋅nηt​ηi​​⋅i+(ηt​ηi​​cosθi​−1−sin2θt​ ​)⋅n​
由Snell’s law公式可得 $\sin {\theta }_t=\frac{{\eta }_i}{{\eta }_t}\cdot \sin {\theta }_i$，
t = η i η t ⋅ i + ( η i η t cos ⁡ θ i − 1 − ( η i η t ) 2 sin ⁡ 2 θ i   ) ⋅ n = η i η t ⋅ i + ( η i η t cos ⁡ θ i − 1 − ( η i η t ) 2 ( 1 − cos ⁡ 2 θ i ) ) ⋅ n

t=ηiηt⋅i+(ηiηtcos⁡θi−1−(ηiηt)2sin2⁡θi )⋅n=ηiηt⋅i+(ηiηtcos⁡θi−1−(ηiηt)2(1−cos2⁡θi))⋅n" role="presentation" style="position: relative;">t==ηiηt⋅i+(ηiηtcosθi−1−(ηiηt)2sin2θi−−−−−−−−−−−−−√ )⋅nηiηt⋅i+(ηiηtcosθi−1−(ηiηt)2(1−cos2θi)−−−−−−−−−−−−−−−−−√)⋅n$$\begin{array}{ccc}\mathbf{t}& =& \frac{{\eta }_i}{{\eta }_t}\cdot \mathbf{i}+\left(\frac{{\eta }_i}{{\eta }_t}\cos {\theta }_i-\sqrt{1-(\frac{{\eta }_i}{{\eta }_t}{)}^2{\sin }^2{\theta }_i}\right)\cdot \mathbf{n}\\ & =& \frac{{\eta }_i}{{\eta }_t}\cdot \mathbf{i}+\left(\frac{{\eta }_i}{{\eta }_t}\cos {\theta }_i-\sqrt{1-(\frac{{\eta }_i}{{\eta }_t}{)}^2(1-{\cos }^2{\theta }_i)}\right)\cdot \mathbf{n}\end{array}$$

t​==​ηt​ηi​​⋅i+⎝⎛​ηt​ηi​​cosθi​−1−(ηt​ηi​​)2sin2θi​ ​ ⎠⎞​⋅nηt​ηi​​⋅i+⎝⎛​ηt​ηi​​cosθi​−1−(ηt​ηi​​)2(1−cos2θi​) ​⎠⎞​⋅n​
其中 $\cos {\theta }_i=\frac{\mathbf{i}\cdot \mathbf{n}}{|\mathbf{i}||\mathbf{n}|}=\mathbf{i}\cdot \mathbf{n}$ 。

全反射现象 Total internal reflection

对于Snell’s law 的公式 $\sin {\theta }_t=\frac{{\eta }_i}{{\eta }_t}\cdot \sin {\theta }_i$ ，当空气中的反射率大于物体内的反射率的时候，有可能会出现 $\frac{{\eta }_i}{{\eta }_t}\cdot \sin {\theta }_i>1$ 的情况。

但是 $\sin {\theta }_t$ 的值一定是小于1的，故等式不可能成立。这个时候光线应该被反射，而不是折射。