<p>A world system is composed of the world lines of observers at rest in the system. The transformation under the constant light speed (TCL), which provides a relativistic transformation between the unique isotropic world system and a rotating world system, can effectively deal with circular motions. Under the unique isotropic frame, the general framework of Mansouri–Sexl (MS) enables us to handle rectilinear motions. Through a derivation based on the MS framework in the standard synchronization, this paper extends the TCL, such that it can also be applied to an arbitrary inertial world system. Circular motions can be considered to be locally and momentarily inertial. Speeds of light in inertial frames are found using the derived TCL via limit operations. It is shown that when a light beam traverses a closed loop in an inertial frame, its average speed is constant, which is consistent with the experiment of Michelson and Morley. Exploiting the TCL, we comprehensively analyze the generalized Sagnac effect in an anisotropic laboratory frame. The analyses include the speeds of light in the rotating frame, not only in the collinear case but also the non-collinear case where the propagation direction of light is not collinear with the direction of fiber motion.</p>