![]() The path taken by the rubber band is geodesic. Imagine a rubber band stretched between two points on a curved surface. Ignoring curvature on a surface is not ideal when you are engineering systems. Finding geodesics is important, as neglecting the curvature introduces errors. A free particle not acted upon by any force excluding gravity follows a curve in curved space-time, which is geodesic. Defining Geodesics as Curves of Zero AccelerationĪ curve on a surface can be said to be geodesic when the acceleration at every point on the curve is either zero or is in parallel to its unit normal. The geodesic or geodesic line is a geometric concept a generalization of the straight line concept in Euclidean geometry for spaces of a more general type. Geodesics refers to the generalized idea of a straight line from a plane to a surface where it forms the shortest distance between two points. Defining Geodesics as Curves of Stationary Length The definition of geodesics can be given in terms of curves of zero acceleration or curves of stationary length. Geodesics define the shortest distance on a surface. Similarly, the parameterization can be based on the energy of the curve, thus the idea of minimizing the energy raises the same questions for geodesics. One parameterization–that the curve not only minimizes the length locally but also be parameterized with constant velocity–can conclude that the distance from f(s) to f(t) is proportional to |s-t| along the geodesic. The method, the calculus of variations, introduces minor technical problems, as there is infinite-dimensional space of different ways to parameterize the short distance between two points. The shortest distance can be determined by writing equations for the length of the curve and using the calculus of variations to minimize the length. In physics or mathematical problems, there are times when we need to find out the shortest path between two points in curved spaces. Geodesics on a sphere is relevant in this context, as it is the shortest path between two points on a spherical surface. It is not easy to calculate the shortest distance on a curved surface. However, as the area becomes larger, it becomes essential to take into account the curvature of the earth to attain accurate calculations. Generally, we take into consideration flat plane surfaces, neglecting the earth’s curvature. Displacement is the shortest distance between two points, which differentiates it from all possible distance measurements. If you ever took physics, you might recall a lesson on the difference between distance and displacement. ![]() The case of geodesics on a sphere is an example of geodesics being closed curves. The geodesic on a sphere is a great circle.
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