Can Cycloid Paths Determine Least Time Courses in 3D?

[Serenie]

when inducing that the cycloid is the least time-taking course between the two points in the two dimension, we have to use calculus of variations.

Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?

 

[blue_leaf77]

Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?

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.