A cycloid is the curve traced by a point on the circumference of a circle as it rolls along a straight line. This mathematical shape is characterized by its distinctive arch-like structure, which consists of a series of peaks and troughs. Cycloids can be generated by rolling a circle of radius r along a flat surface, producing a smooth, wave-like pattern. Cycloids have applications in various fields, including physics and engineering. They are important in the study of mechanics and calculus, particularly in problems involving the motion of objects. The cycloid is also known for its unique properties, such as being the solution to the brachistochrone problem, which seeks the fastest path between two points under the influence of gravity.