Reflection and Refraction

Recursive Ray Tracing

At an intersection point, we can spawn additional rays:

• Reflection rays for mirrors
• Refraction rays for transparent materials

Reflection

For mirror-like surfaces, a reflected ray is spawned:

$\mathbf{R} = \mathbf{D} - 2(\mathbf{D} \cdot \mathbf{N})\mathbf{N}$

where $\mathbf{D}$ is the incoming ray direction and $\mathbf{N}$ is the surface normal.

Reflection Direction Derivation

The reflection is the incident direction mirrored about the normal:

• Component parallel to $\mathbf{N}$: reversed
• Component perpendicular to $\mathbf{N}$: unchanged

Refraction (Transparency)

Transparent objects bend light according to Snell’s law:

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

where $n_1$ and $n_2$ are refractive indices.

Refracted Direction

$\mathbf{T} = \frac{n_1}{n_2}\mathbf{D} + \left(\frac{n_1}{n_2}\cos\theta_1 - \cos\theta_2\right)\mathbf{N}$

Total Internal Reflection

When light travels from a denser to less dense medium:

• If incident angle exceeds critical angle: No refraction, only reflection
• Critical angle: $\theta_c = \arcsin(n_2/n_1)$

Fresnel Effect

At glancing angles, more light is reflected; at normal incidence, more is refracted:

• Approximate with Schlick’s approximation
• Important for realistic glass and water

Combining Effects

The final colour combines:

• Local shading (Phong)
• Reflected colour (weighted by reflectivity)
• Refracted colour (weighted by transparency)

$I = I_{\text{local}} + k_r I_{\text{reflected}} + k_t I_{\text{refracted}}$

Summary

• Reflection rays use the mirror direction formula
• Refraction follows Snell’s law
• Total internal reflection occurs above the critical angle
• Combined colour blends local, reflected, and refracted contributions