Geodesic Haversine spherical arc derivation vs flat Mercator linear chord error

When flight paths and orbital ground tracks are rendered as straight lines on standard Web Mercator (EPSG:3857) maps, they deceive human perception. Web Mercator is conformal—preserving local angles at the expense of severe polar areal distortion ($\sec^2\phi \to \infty$).

A straight line drawn between two coordinates on a Mercator map is a Rhumb line (loxodrome), not the shortest distance on Earth. Spacecraft orbital trajectories follow great circles governed by the Riemannian 2-sphere metric $ds^2 = R^2(d\phi^2 + \cos^2\phi\,d\lambda^2)$.

1. Haversine Central Angular Distance & 3D Slerp

Given origin $P_1(\phi_1, \lambda_1)$ and destination $P_2(\phi_2, \lambda_2)$, central angle $\Delta\sigma$ is calculated via:

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

Intermediate surface waypoints at fraction $f \in [0, 1]$ are interpolated via 3D unit vectors $\mathbf{v}_1, \mathbf{v}_2 \in \mathbb{S}^2$:

$\mathbf{v}(f) = \frac{\sin((1 - f)\Delta\sigma)}{\sin\Delta\sigma}\mathbf{v}_1 + \frac{\sin(f\Delta\sigma)}{\sin\Delta\sigma}\mathbf{v}_2$

2. Empirical Divergence on the London → Tokyo Corridor

On the sub-polar flight path from London (51.51° N) to Tokyo (35.68° N):

Metric	Great-Circle Geodesic	Flat Mercator Linear Chord	Distortion Delta
Path Distance	5,939.48 miles	7,017.56 miles	+1,078.08 miles (+18.15%)
Max North Latitude	68.99° N (Arctic Siberia)	51.51° N (Kazakhstan)	1,857 miles lateral separation