Continuous unfurl engine: Pythagorean limb departure and dynamic horizon clipping

When transitioning between a 3D orthographic sphere and a planar cylindrical projection, standard GIS frameworks perform hard cuts or opacity fades. GeoGenesis implements a frame-by-frame continuous geometric morpher driven by the Pythagorean limb departure envelope $\sqrt{R^2 - y^2}$.

1. Parallel Limb Departure Derivation

On an orthographic sphere of radius $R$, surface coordinates $(\lambda, \phi)$ map to Cartesian screen coordinates:

$x_{\text{ortho}}(\lambda, \phi) = R \cos\phi \sin\lambda, \quad y_{\text{ortho}}(\lambda, \phi) = R \sin\phi$

At vertical offset $y \in [-R, R]$, latitude $\phi$ is fixed by $\sin\phi = y / R$. Applying the Pythagorean theorem to the cross-sectional circle parallel to the equatorial plane at height $y$:

$x_{\text{limb}}^2 + y^2 = R^2 \implies x_{\text{limb}}(y) = \pm \sqrt{R^2 - y^2} = \pm R \cos\phi$

2. Continuous Morphing Topology Function

For unfurl parameter $t \in [0, 1]$ ($t = 0$: sphere; $t = 1$: planar projection $P$), the blended transformation $\mathcal{T}_t$ is defined as:

$\mathcal{T}_t(\lambda, \phi; P) = \begin{bmatrix} (1 - t) \cdot R \cos\phi \sin\lambda + t \cdot X_P(\lambda, \phi) \\ (1 - t) \cdot R \sin\phi + t \cdot Y_P(\lambda, \phi) \end{bmatrix}$

3. Dynamic Horizon Clipping $\alpha(t)$

At $t = 0$, only the visible hemisphere facing the camera is rendered ($\alpha_0 = 90^\circ = \pi/2\text{ rad}$). As $t \to 1$, the clipping boundary expands to reveal the complete $360^\circ$ planet:

$\alpha(t) = \frac{\pi}{2} (1 + t)$

Antimeridian cutting (d3.geoClipAntimeridian) activates when $t > 0.001$, preventing meridian seam tearing during planar layouts while preserving continuous 3D rotation at $t = 0$.