(a) Coordinate systems and corresponding coordinate rotations based on TCS and NCS in four equal-sided nonplanar ring resonators (NPRO), : folding angle, the cavity length of all four sides are equal and the total cavity length is , and : spherical mirrors with radius of and , and : planar mirrors, : incident angles on four mirrors, : terminal points of the resonator, , , , , , : the midpoints of straight lines , , , , , and separately, and : coordinate rotation angles based on TCS and NCS respectively; (b) one single segment of a general resonator, : the reflection points, and : coordinate rotation angles based on TCS and NCS respectively, : facular (transverse section) of the incident beam before being reflected from points ; (c) schematic diagram of experimental result on optical-axis perturbation caused by spherical mirror’s radial displacements in square ring resonators (SRR), SRR: a special case of NPRO in the case of and , : radial displacements of with the same value of , and : the positions of points after perturbations of by referring to TCS and NCS, respectively. The ideal optical-axes and the real optical-axes (after special perturbations) are represented by blue solid lines and red dashed lines, respectively. Spherical mirror’s positions, after radial displacements, are illustrated with red solid arcs. : the binormals at points , and : coordinate systems for the incident beam (based on TCS and NCS, respectively) before being reflected from points , and : coordinate systems for the reflected beam (based on TCS and NCS, respectively) after being reflected from points . (Note: The positive directions of and are along the directions of ; the positive directions of and are along the direction of beam propagation; , , and are located at the incident planes of , , , and separately).