Geodesic — Definition, Formula & Examples

A geodesic is the shortest or locally shortest path between two points on a curved surface or in a curved space. On a sphere, geodesics are great circles; on a flat plane, they are straight lines.

A geodesic on a Riemannian manifold $(M, g)$ is a curve $\gamma(t)$ whose tangent vector is parallel-transported along itself, satisfying the geodesic equation $\frac{d^2 x^\mu}{dt^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt} = 0$ , where $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols of the metric $g$ . Equivalently, a geodesic is a curve that locally extremizes arc length between its endpoints.

Key Formula

\frac{d^2 x^\mu}{dt^2} + \Gamma^\mu_{\alpha\beta}\,\frac{dx^\alpha}{dt}\,\frac{dx^\beta}{dt} = 0

Where:

$x^\mu(t)$ = Coordinates of the curve parameterized by t
$\Gamma^\mu_{\alpha\beta}$ = Christoffel symbols of the Levi-Civita connection, computed from the metric tensor
$t$ = Affine parameter along the curve (often arc length)

How It Works

To find a geodesic, you solve the geodesic equation, a system of second-order ordinary differential equations involving the Christoffel symbols of the metric. The Christoffel symbols encode how the coordinate basis vectors change from point to point on the manifold. Given initial position and velocity, the geodesic equation produces a unique curve. On simple surfaces like spheres, symmetry arguments can identify geodesics without solving the full equation.

Worked Example

Problem: Find the geodesic (shortest path) on a sphere of radius R = 6371 km between the North Pole (90°N, 0°E) and a point on the equator (0°N, 30°E).

Identify the geodesic type: On a sphere, all geodesics are great circles — circles whose center coincides with the center of the sphere.

Find the great-circle arc: The angular separation between the North Pole (latitude 90°) and any equatorial point is exactly 90°, regardless of longitude. Convert to radians.

\theta = 90° = \frac{\pi}{2} \text{ radians}

Compute the geodesic distance: The arc length along a great circle is the radius times the central angle.

d = R\,\theta = 6371 \times \frac{\pi}{2} \approx 10{,}008 \text{ km}

Answer: The geodesic is the great-circle arc from the North Pole through the 30°E meridian to the equator, with length approximately 10,008 km.

Why It Matters

Geodesics are central to general relativity, where freely falling objects follow geodesics in curved spacetime. Navigation and aviation use great-circle geodesics to compute fuel-efficient flight paths. In computer graphics and mesh processing, geodesic distances on surfaces guide texture mapping, remeshing, and shortest-path queries.

Common Mistakes

Mistake: Assuming a geodesic is always the global shortest path between two points.

Correction: A geodesic is only locally length-minimizing. On a sphere, the longer arc of a great circle is also a geodesic but is not the shortest path. Geodesics extremize length; they need not minimize it.