Geodesics on Surfaces

The shortest paths along curved surfaces with animated particle tracing

Geodesics

Walk across a curved surface without ever turning left or right, and you trace a geodesic: a straight line on a plane, a great circle on a sphere, and something stranger everywhere else.

Geodesic Particle Tracer

Pick a surface and watch the glowing particle go straight ahead. Its trail shows how a path with no turning still curves through space.

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.

Go deeper: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.

Go deeper: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.

Geodesic Spray

Fire geodesics in every direction from one point. On the sphere they focus back together; on the saddle they fly apart.

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.

Go deeper: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.

Geodesic vs Straight Line

Compare the shortest path on the surface with the straight line through 3D space. The geodesic is always longer, because it has to stay on the surface.

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

Closed Geodesics Hunter

Can you find a closed geodesic, one that returns to where it started? Nudge the direction angle to shrink the closure gap; tiny changes matter.

Closure Gap: 1.8186

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

Key Takeaways

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

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