Blinn-Phong specular ocean glint and raytraced atmospheric scattering

TerraGemini renders oceanic reflection and planetary atmosphere using a raytraced billboard quad covering the screen-space bounding disc of the planet ($u^2 + v^2 \le 1.0$).

1. Spherical Normal Reconstruction

For any fragment $(u, v) \in [-1, 1]^2$, fragments outside $r^2 = u^2 + v^2 \le 1.0$ are discarded. The unit normal $\mathbf{N}$ on the sphere surface is derived analytically:

$\mathbf{N} = \begin{pmatrix} u \\ -v \\ \sqrt{1.0 - r^2} \end{pmatrix}$

2. Solar Vector & Half-Angle Glint

With solar azimuth $\theta_{\text{sun}}$ and constant solar elevation $\alpha = 0.35\text{ rad}$, the normalized light vector $\mathbf{L}$ is constructed. Blinn-Phong specular intensity is calculated using the half-angle vector $\mathbf{H} = \frac{\mathbf{L} + \mathbf{V}}{\|\mathbf{L} + \mathbf{V}\|}$ with an exponent of $s = 38.0$:

$I_{\text{spec}} = (\max(\mathbf{N} \cdot \mathbf{H}, 0.0))^{38} \cdot \text{step}(0.001, \mathbf{N} \cdot \mathbf{L})$

The step function guarantees that the unlit hemisphere receives zero specular highlight.

3. Rayleigh Atmospheric Limb & Composite

Limb scattering $I_{\text{limb}} = (1.0 - \sqrt{1.0 - r^2})^{2.6}$ produces an atmospheric rim halo. The final composited fragment combines diffuse ocean tone, specular glint, and atmospheric scattering in a single pass:

$\mathbf{C}_{\text{final}} = \mathbf{C}_{\text{ocean}} \cdot (0.18 + 0.82(\mathbf{N} \cdot \mathbf{L})) + \mathbf{C}_{\text{specular}} \cdot (1.85 \cdot I_{\text{spec}}) + \mathbf{C}_{\text{atmos}} \cdot (0.75 \cdot I_{\text{limb}})$