Geodesics on Surfaces

The shortest paths along curved surfaces with animated particle tracing

Geodesics

A geodesic is the “straightest possible” curve on a surface — the path a particle would follow if it experienced no forces other than the constraint of staying on the surface.

On a sphere, geodesics are great circles. On a plane, they're straight lines. On more complex surfaces, they curve in fascinating ways dictated by the geometry.

Interactive: Geodesic Particle Tracer

Watch a glowing particle trace a geodesic path. The trail shows the history of the particle's journey — always following the straightest possible path on the curved surface.

Watch the glowing particle trace a geodesic — the shortest path between points on the surface. On a sphere, geodesics are great circles. On other surfaces, they curve to follow the geometry.

The Geodesic Equation

Geodesics satisfy a system of differential equations involving the Christoffel symbols Γ:

d²uk/dt² + Γkij (dui/dt)(duj/dt) = 0

This equation says that the “acceleration” in the surface coordinates must exactly cancel the curvature effects, resulting in zero geodesic curvature.

Interactive: Geodesic Spray

Fire geodesics in all directions from a single point. Watch how they spread out, focus, or meet again depending on the surface's curvature.

Geodesics fired in all directions from a single point create a “spray” pattern. On a sphere, they all meet again at the antipodal point. On other surfaces, they diverge or focus based on the curvature.

How Curvature Affects Geodesics

Positive Curvature (K > 0)

Geodesics converge and eventually meet. On a sphere, all geodesics from a point meet at the antipodal point.

Zero Curvature (K = 0)

Geodesics behave like straight lines — they neither converge nor diverge. This includes planes and cylinders.

Negative Curvature (K < 0)

Geodesics diverge exponentially. On a saddle, nearby geodesics quickly separate.

Interactive: Geodesic vs Straight Line

Compare the geodesic path (the shortest path on the surface) with the straight line through 3D space. The geodesic is always longer but stays on the surface!

Geodesic Length
1.990
Straight Line (3D)
1.649
Ratio
1.206x
Geodesic (on surface)
Straight line (through space)
Start
End

Interactive: Closed Geodesics Hunter

Can you find a closed geodesic — one that returns to its starting point? Adjust the direction angle to minimize the closure gap.

Closure Gap: 1.8186

All geodesics on a sphere are great circles (closed!)

Key Properties of Geodesics

  • •Locally shortest: Between nearby points, a geodesic is the shortest path on the surface
  • •Zero geodesic curvature: The curvature vector is always normal to the surface (no “turning” within the surface)
  • •Constant speed: A geodesic parametrized by arc length maintains constant velocity magnitude
  • •Intrinsic: Geodesics depend only on the metric, not the embedding in 3D space

Key Takeaways

  • Geodesics are the straightest curves on a surface (zero geodesic curvature)
  • They satisfy the geodesic equation involving Christoffel symbols
  • On a sphere, geodesics are great circles; on a plane, straight lines
  • Positive curvature makes geodesics converge; negative curvature makes them diverge
  • Closed geodesics are special — they return to their starting point

Next: Parallel Transport — moving vectors along curves while keeping them “as parallel as possible” on a curved surface.