Cycloid

Applications:

Pendulum Design:  The cycloidal path is used in the design of pendulums for clocks to achieve isochronous (equal time) oscillations, improving accuracy.

Brachistochrone Problem: The cycloid is the solution to the brachistochrone problem, representing the curve of the fastest descent between two points under gravity.

Gear Design: Cycloidal curves are used in the design of gear teeth to minimize friction and wear, particularly in high-precision mechanisms like watches.

Roller Coaster Loops: Cycloidal loops can be used in roller coaster design for smoother and more efficient transitions in loop-the-loop sections.

Cycloidal Drives: Cycloidal drives, a type of gear mechanism, use cycloid curves to achieve high torque and smooth operation in compact designs.

Robotics: In robotics, cycloidal paths are used in trajectory planning for smooth and efficient movement.