Geodésica

Geodésica refers to the shortest path between two points on a curved surface. On a flat plane, this shortest path is a straight line. However, on a sphere or other curved surfaces, the shortest path follows the curvature of the surface. This concept is fundamental in geometry and has significant applications in various fields, including navigation, geodesy, and general relativity.

In spherical geometry, the geodésica between two points is an arc of a great circle. A great circle is the intersection of the sphere with a plane that passes through the center of the sphere. For example, lines of longitude on Earth are segments of great circles, and the shortest distance between two points on the Earth's surface can be approximated by following the arc of a great circle connecting them.

The mathematical study of geodesics is a core part of differential geometry. The equation describing a geodesic can be derived using the calculus of variations. In the context of general relativity, spacetime is treated as a curved manifold, and the paths of objects moving freely under gravity are described as geodesics within this spacetime. This means that gravity is not a force in the traditional sense, but rather a manifestation of the curvature of spacetime. The concept also extends to higher-dimensional spaces and more complex geometries.