The discussion confirms that cycloid paths represent the least time-taking courses between two points in three dimensions, provided the points lie within a plane that includes the gravitational force vector. The application of calculus of variations is essential for deriving this principle. The cycloid solution is universally applicable in 3D scenarios, ensuring that a cycloidal trajectory can always be established for any given pair of points.
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In principle yes, in fact the cycloid solution applies to a 3D curve when the two points lie in a plane that also contains the gravitational force vector. Since such plane always exists for any given pair of points, the trajectory will always be a cycloid.Serenie said:Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?