Geodesic

Klein quartic with closed geodesics

Transpolar geodesic on a triaxial ellipsoid case A

Insect on a torus tracing out a non-trivial geodesic

Spherical triangle

End of universe

Geodesic refers to the shortest path between two points on a curved surface, such as the surface of the Earth or a sphere. The concept of geodesics is fundamental in various fields, including mathematics, physics, and geography. In the context of mathematics, particularly in differential geometry, a geodesic is a curve that is everywhere locally a distance minimizer. These paths are not necessarily straight lines due to the curvature of the space in which they are embedded.

Definition

In a more formal mathematical setting, a geodesic is defined with respect to a Riemannian manifold or a more general Finsler manifold, which are spaces where a notion of distance (derived from a metric tensor) is defined. A geodesic in these manifolds is a curve whose tangent vectors remain parallel if they are transported along it. If the manifold is equipped with a metric tensor, geodesics are the paths that minimize the distance between points, which is calculated using this tensor.

Geodesics in General Relativity

In the field of general relativity, geodesics take on a significant role. General relativity, a theory of gravitation developed by Albert Einstein, describes gravity not as a force between masses but as a result of the curvature of spacetime caused by mass-energy. In this framework, objects in free fall follow paths in spacetime that are geodesics of the spacetime geometry. This concept explains why, for example, planets orbit stars in elliptical paths, as they are following geodesic paths in the curved spacetime around the star.

Geodesic Equations

The equations that describe geodesics can be derived from the Euler-Lagrange equations in the calculus of variations. For a Riemannian manifold, these equations take the form of second-order differential equations that describe how the coordinates of a point on a geodesic curve change with respect to a parameter (often taken to be the arc length of the curve).

Applications

Geodesics have applications in various fields: - In geography and geodesy, the concept of geodesics is used to define the shortest route between two points on the Earth's surface, taking into account its curvature. This is crucial for navigation and mapmaking. - In computer graphics and modeling, geodesic algorithms are used to calculate the shortest paths on the surface of a model, which can be important for texture mapping, mesh simplification, and other applications. - In physics, beyond general relativity, geodesics are used in the study of geometric optics and quantum field theory.

See Also

• Riemannian manifold
• General relativity
• Differential geometry
• Euler-Lagrange equations

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