The shortest distance between two points is always a straight line in Euclidean geometry".
This is the geometry that is usually learned in school, where the figures are two-dimensional and represented on a flat surface like a notebook sheet.
In real life, the shortest distance is a curve called geodesic. That's because our planet is not flat! Thus, Euclidean geometry is not used, but Riemanian geometry.
This is the concept that flight planners use to chart airplane routes in order to save time and fuel. From a practical point of view, in most cases, geodesic is the shortest curve that joins two points.
This effect has interesting implications; for example, when you fly on an airplane, the path it takes to go from one destination to another does not follow a "straight line", as many people imagine. It follows the “curvature” of the Earth, making small adjustments in the direction of travel, in order to cover the shortest possible stretch. If the plane were simply "in a straight line", it would end up traveling a longer trajectory than it does when following the land curvature.
Riemannian geometry is a branch of mathematics that studies curved spaces. It was developed by Bernhard Riemann in the 19th century as a generalization of Euclidean geometry, which describes flat surfaces like a tabletop. In Euclidean geometry, the shortest distance between two points is a straight line, and angles always add up in predictable ways. But in Riemannian geometry, space can be curved, meaning that the usual rules about distances and angles don’t always apply.
A key idea in Riemannian geometry is the concept of a manifold, which is a space that looks flat when examined up close but may be curved when viewed as a whole. A good example is the surface of the Earth. If you’re standing on the ground, it seems flat, but when you look at it from space, you can see it is actually curved like a sphere. Riemannian geometry provides a way to measure distances, angles, and curvature in these kinds of spaces using a mathematical tool called a metric tensor, which assigns a way to calculate distances at every point in the space.
One of the most famous applications of Riemannian geometry is in Albert Einstein’s theory of General Relativity. Einstein proposed that gravity isn’t just a force pulling objects together but rather a bending of space and time caused by massive objects like planets and stars. The mathematical framework used to describe this curvature is based on Riemannian geometry. In essence, it helps us understand how objects move in curved space, whether they are planets orbiting a star or light bending around a black hole.
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