Invariant of circular surface with fixed radius and its application in triad

The invariants of circular surface with fixed radius are investigated by means of differential geometry. Two moving frames are established, which are respectively on the central curve and the circle generatrix of circular surface. Based on the unit normal vector of the circular plane and the arc length of its spherical image curve, the former frame and the central curve are combined and the vector equation of circular surface is derived and some properties of circular surface are revealed to judge whether a surface to be a circular surface. The latter frame is built on the circle generatrix and the trajectory of its origin is called the spine curve of circular surface. Through the differential operation of the frame, the kinematic geometric properties of circular surface are studied. The kinematic invariants are derived and their geometric meanings are presented. Some degenerative circular surfaces are studied, such as torus, cylindrical spiral circular surface and spherical surface. The sufficient and necessary conditions are given for a circular surface to be one of them. Then constraint circular surface for a triad (three-link chain) with cylindrical, revolute and spherical joints is discussed.