This simulator allows you to numerically investigate the classical brachistochrone problem: determining the trajectory along which a particle, subject only to uniform gravity and starting from rest, takes the least time to travel from point A to point B.

Surprisingly, the path of least time is neither a straight line nor a circular arc, but rather a cycloid. Mathematically, the brachistochrone can be described by the parametric equations

$ x(\theta) = R(\theta - \sin\theta) $ $ y(\theta) = R(1 - \cos\theta), $

where $R$ is the generating radius of the cycloid and $\theta$ is the parameter that describes the motion along the curve. This form arises from minimizing the descent time using the calculus of variations.

In the simulator, the initial point A is fixed, while the final point B can be freely chosen by the user. The trajectory between A and B can assume different shapes, allowing comparison of the descent time in each case.

How to use the simulator:

• Drag point B directly on the screen to change the particle's final position.
• Adjust the shape of the trajectory by moving the green control points (1 and 2), which modify the curve between A and B.
• Check the "Brachistochrone" option to display, as a red dashed line, the theoretical curve of minimum time. This line serves as a visual reference so that you can compare and adjust your trajectory.

Explore different configurations and observe how small changes in the shape of the curve can alter the descent time.