Geometry lab
Geometry & Invariants
Fly straight on a curved surface. Reveal the local basis and watch the components change while the tangent remains intrinsically parallel.
Read “Geometry Is Everywhere”— Great circle · → T, unit tangent · → North · → East
Drag with mouse or touch to rotate. With the globe focused, use arrow keys. Faint dashed lines lie on the far side. The triangular marker shows flight direction.
Follow the flight
Ready. Apply a route or press Start.
One vector, changing components
T = Tᴺ eᴺ + Tᴱ eᴱ in the orthonormal north/east frame. These are not latitude/longitude coordinate components.
Compare ordinary component derivatives dTᴺ/ds, dTᴱ/ds with the covariant derivative DT/ds = 0. Here s is distance in km.
The moving frame contributes Ω = sin(φ) dλ/ds, with latitude φ and longitude λ. Thus (DT/ds)ᴺ = dTᴺ/ds + Ω Tᴱ = 0 and (DT/ds)ᴱ = dTᴱ/ds − Ω Tᴺ = 0. These are orthonormal-frame connection terms, not coordinate Christoffel symbols.
Parallel does not mean fixed in 3D
Small gold arrows are transported tangents along the same geodesic. For the Levi-Civita connection on a sphere, the tangential projection of dT/ds vanishes: DT/ds = 0.
In the surrounding Cartesian space, dT/ds = −r̂/R. The pink radial arrow shows its direction (rescaled for visibility). The vector T itself rotates:
The same route on a flat map
Equirectangular projection: horizontal position is longitude, vertical position is latitude. The map cuts at ±180°; a break there is not a break in the flight. Except for special routes, a great circle appears curved. The poles stretch into entire horizontal edges. Grid only; no geographic basemap is loaded.