Parallel Transport on a Sphere 3D

The idea

On a flat sheet you can carry an arrow around a closed loop and it comes back pointing the same way. On a sphere it does not. Parallel transport means moving a tangent vector so that it is never turned relative to the surface: along a geodesic (great circle) it keeps a constant angle to the path. After a closed loop the vector comes back rotated by the holonomy angle.

Gauss-Bonnet

For a region of a surface with Gaussian curvature K bounded by geodesic arcs, sum of exterior angles + integral of K dA = 2 pi. On a sphere of radius R, K = 1/R^2, so the holonomy angle of a counter-clockwise loop is area / R^2 radians. For a geodesic triangle with interior angles A, B, C this is the spherical excess E = A + B + C - pi (Girard, 1629). A triangle with three right angles (an octant) has excess pi/2 and the vector returns turned by exactly 90 degrees.

How the page computes it

The path is cut into small steps. At each step the vector is rotated by the rotation that carries the point p_k to p_{k+1} (axis p_k x p_{k+1}); on a geodesic this is exact. The transported vector is compared with the start vector at the end: holonomy = atan2(n . (v0 x v), v0 . v). For a latitude circle a second, independent method integrates dv/dt = -(v . dn/dt) n with fourth-order Runge-Kutta; the chord-step method converges with second order, RK4 much faster (see the error graph). The exact value is 2 pi (1 - sin lat), the area of the polar cap divided by R^2.

Foucault pendulum

A Foucault pendulum at latitude lat keeps its swing plane parallel-transported along the latitude circle that the Earth carries it round. Seen from the ground its plane turns 360 sin(lat) degrees per sidereal day, that is 15 sin(lat) degrees per hour: 0 at the equator, a full turn per day at the poles. This equals 360 degrees minus the holonomy of the circle, because the ground frame itself turns by 360 degrees around the circle.

Honest simplifications

The sphere is perfect (Earth is slightly oblate, which changes holonomy by about 0.3 %). Triangles are minor triangles with sides shorter than 180 degrees. Vertices at latitude 90 degrees are moved to 89.999 degrees to avoid a degenerate longitude. Foucault values ignore air drag and the pendulum's finite size.

Independent checks on this page

• Transport rotation of a skew triangle versus the angle-sum excess and the Van Oosterom-Strackee solid-angle formula.
• A 400,000-point Monte-Carlo estimate of the triangle's area on the sphere versus the excess.
• RK4 integration of the transport ODE versus the closed form 2 pi (1 - sin lat) at latitude 52 degrees.

What the 3D scene adds

A Blender-built marble globe stands in a small gallery with a brass meridian ring. The path is a glowing tube on its surface, the walker carries a physical arrow rod that turns exactly as the transported vector, and the enclosed region is a translucent cap built from the same loop as the 2D page.