Abstract: The physical process of the formation of a cycloid is analyzed and the parameter equation is obtained. According to the law of vector for the velocity,the direction of tangent of each point of the cycloid and the length of the cycloid are calculated. An analogy on the Fermat＇s theory and the equivalence of simple harmonic motion and circular motion are made and thus the fastest falling line and ＂isochronous＂of the cycloid are proved.

Key words: cycloid, brachistochrone, tautochrone