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In mathematics, geodesics are studied in the field of differential geometry and play a crucial role in various areas, including physics and computer graphics. This Wikipedia page provides an overview of geodesics, discussing their definition, properties, and applications. The page starts by introducing the concept of geodesics in differential geometry, explaining their relationship with the metric tensor and the notion of curvature. It then delves into the mathematical formalism used to define geodesics on different types of surfaces, such as Euclidean space, spheres, and hyperbolic planes. The properties of geodesics are explored, including their parametrization, equations of motion, and characterization as extremal curves. The page also covers geodesic distance, which measures the length of a geodesic arc between two points, and presents different methods for calculating it. Furthermore, the Wikipedia page discusses various applications of geodesics across different disciplines. In physics, geodesics are integral to understanding the motion of particles in gravitational fields, as they represent the paths that gravitational bodies follow. Geodesics are also employed in computer graphics for tasks like designing 3D models and animating characters, where they are utilized to create realistic movements and simulations. The page concludes with a section on notable examples of geodesics, such as the great circles on the Earth's surface and the geodesic domes designed by architect Buckminster Fuller. It also mentions how geodesic concepts have been extended and adapted in different contexts, such as geodesic lines in graph theory and geodesic paths in network analysis. Overall, the Wikipedia page on geodesics provides a comprehensive overview of this fundamental concept in mathematics and its applications in various fields.