A geodesic on a surface generalizes a straight line in Euclidean geometry: For sufficiently-nearby points on a surface, a geodesic joining them is the shortest path between them and lying in the surface. Unlike in Euclidean geometry, a geodesic generally "winds around" on a surface, and is not necessarily the shortest path between two points on the geodesic.
Given a point on a surface and a tangent vector at that point, there is a unique geodesic starting at the given point with given initial velocity. The image shows an ellipsoid with three unequal axes, and geodesics radiating from the point at upper right, with the color of each geodesic grading as the initial velocity sweeps a circle. The front of the ellipsoid is lit, the back lies in shadow. The chaotic nature of geodesics on an ellipsoid can be glimpsed, as can "focusing" behavior where "nearby geodesics nearly come together."
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