Mathematical Explanation of the Heightfield Ocean Shader

1. Definition of the Ocean Surface (Heightfield)

The water surface is defined as a scalar function on the x-z plane:

y = h(x,z,t)

In this shader, several sine waves are superimposed:

h(x,z,t) = Σ A_i · sin( k_i · (x,z) + ω_i t )

where A_i is the amplitude, k_i the wave vector, and ω_i the time-dependent frequency.

2. Ray Parametric Equation

With camera position r₀ and ray direction d, the ray is:

r(t) = r₀ + td,   t ≥ 0

The y-component is:

y_ray(t) = r_0y + t d_y

3. Intersection Condition

The intersection occurs when:

y_ray(t) = h(x_ray(t), z_ray(t), t_global)

Equivalently, solving the root-finding equation:

f(t) = (r_0y + t d_y) − h(r_0x+t d_x, r_0z+t d_z, t_global) = 0

4. Numerical Root-Finding

1. Bracket search: Step forward exponentially until f(t) changes sign, i.e. f(t_a)·f(t_b) < 0.
2. Hybrid Secant + Bisection:
   t_n+1 = t_b − f(t_b) (t_b − t_a) / (f(t_b) − f(t_a))

   If unstable, fallback to bisection:

   t_n+1 = (t_a + t_b) / 2

The converged solution t* gives the intersection with the water surface.

5. Surface Normal

The gradient of the heightfield gives the normal:

∇h(x,z) = ( ∂h/∂x,   1,   ∂h/∂z )

Normalized:

n = (−h_x, 1, −h_z) / √(h_x² + h_z² + 1)

6. Shading Model

• Diffuse: L_diff ∝ max( n·l, 0 )
• Specular (Blinn-Phong): L_spec ∝ (max(n·h,0))^p
• Fresnel (Schlick): F(θ) = F₀ + (1 − F₀)(1 − cosθ)⁵
• Reflection: r = d − 2(n·d)n
• Fog: C = e^{−α t*}C_water + (1−e^{−α t*})C_sky

In summary: The shader formulates the water intersection as a 1D root-finding problem f(t)=0, solves it numerically, computes the surface normal, and then applies a physically motivated shading model combining diffuse, specular, Fresnel reflection, foam, and distance fog.