Multi-projection topology mutators: Taylor sinc guards and Newton-Raphson Kepler solvers

GeoGenesis supports four cartographic projection topologies: Orthographic (3D disk), Equirectangular (Plate Carrée), Winkel Tripel (National Geographic standard), and Mollweide (equal-area ellipse).

1. Winkel Tripel: 4th-Order Taylor Series Sinc Guard

Winkel Tripel is calculated as the arithmetic mean of the Equirectangular and Aitoff projections, with standard parallel $\phi_1 = \arccos(2/\pi) \approx 50.467^\circ$:

$\alpha = \arccos\left(\cos\phi \cos\frac{\lambda}{2}\right)$

At the projection center ($\lambda \to 0, \phi \to 0 \implies \alpha \to 0$), direct evaluation of $\sin(\alpha)/\alpha$ causes floating-point division by zero. A 4th-order Taylor series expansion eliminates the singularity:

$\text{sinc}(\alpha) = \begin{cases} 1 - \frac{\alpha^2}{6} + \frac{\alpha^4}{120} & \text{if } |\alpha| < 10^{-6} \\ \frac{\sin\alpha}{\alpha} & \text{otherwise} \end{cases}$

$X_{\text{wintri}}(\lambda, \phi) = \frac{R}{2} \left[ \lambda \cdot \frac{2}{\pi} + \frac{2 \cos\phi \sin(\lambda / 2)}{\text{sinc}(\alpha)} \right], \quad Y_{\text{wintri}}(\lambda, \phi) = \frac{R}{2} \left[ \phi + \frac{\sin\phi}{\text{sinc}(\alpha)} \right]$

2. Mollweide Equal-Area: Newton-Raphson Kepler Solver

The Mollweide projection requires solving the transcendental Kepler equation for parametric latitude $\theta$:

$2\theta + \sin(2\theta) = \pi \sin\phi$

Using Newton-Raphson iteration seeded with $\theta_0 = \phi$:

$\theta_{n+1} = \theta_n - \frac{2\theta_n + \sin(2\theta_n) - \pi \sin\phi}{2 + 2\cos(2\theta_n)}$

The solver terminates when $|\Delta\theta| < 10^{-11}$ (typically $\le 4$ iterations).