Geodesic arc morphing: spherical SLERP flight paths across dynamic projections

Great-circle paths represent the shortest distance between two points on a spherical surface. As a map morphs between 3D spherical and 2D planar topologies, these trajectories must warp dynamically without distortion or seam breaks.

1. Great-Circle Distance & SLERP Interpolation

Given origin $P_1 = (\lambda_1, \phi_1)$ and destination $P_2 = (\lambda_2, \phi_2)$, unit surface vectors $\mathbf{v}_1, \mathbf{v}_2 \in \mathbb{S}^2$ are computed. Central angle $\Omega$ is obtained via the Haversine equation:

$h = \sin^2\left(\frac{\Delta\phi}{2}\right) + \cos\phi_1 \cos\phi_2 \sin^2\left(\frac{\Delta\lambda}{2}\right), \quad \Omega = 2 \arcsin(\min(1, \sqrt{h}))$

For path parameter $u \in [0, 1]$ sampled across 64 waypoints, spherical linear interpolation (SLERP) generates intermediate surface unit vectors:

$\mathbf{v}(u) = \frac{\sin((1 - u)\Omega)}{\sin\Omega} \mathbf{v}_1 + \frac{\sin(u\Omega)}{\sin\Omega} \mathbf{v}_2$

2. Continuous Projection Morphing

Each waypoint $(\lambda(u), \phi(u))$ is transformed through the continuous projection operator:

$\mathbf{p}_{\text{screen}}(u, t) = \mathcal{T}_t(\lambda(u), \phi(u); P)$

On the 3D globe ($t < 0.5$), back-face culling is enforced using camera normal dot products ($\mathbf{v}(u) \cdot \mathbf{n}_{\text{camera}} > 0$). Trajectories are rendered with a phosphor glow halo and an animated photon particle with trail decay.