Geodetics

Geodetics are the shortest paths between two points on a curved surface, most commonly the surface of the Earth. Unlike straight lines in Euclidean geometry, geodetics account for the curvature of the space they traverse. On a sphere, the geodetic is a segment of a great circle, which is a circle on the sphere whose center coincides with the center of the sphere. For example, the shortest flight path between two cities often follows a geodetic.

The concept of geodetics is fundamental in geodesy, the science of measuring and understanding the Earth's geometric shape, orientation in space, and gravitational field. Geodetic measurements are crucial for navigation, mapping, and satellite operations. The shape of the Earth is not a perfect sphere but an oblate spheroid, meaning it bulges at the equator. Therefore, geodetic calculations for the Earth involve more complex mathematical formulas than simple spherical geometry.

In differential geometry, geodetics are defined as curves whose tangent vectors remain parallel when parallel transported along the curve. This means that the curve has no intrinsic curvature. In simpler terms, if you were to imagine a tiny ant walking along a geodetic on a surface, it would feel like it was walking in a straight line, even though the surface itself is curved. The study of geodetics is essential for understanding the geometry of curved spaces.