The electron spin resonance lineshape (ESRLS) function for the electron spin resonance linewidth (ESRLW) of $Cr^{3+}$ (S = 3/2) in ferroelectric lithium niobate single crystals doped with 0.05 wt% of Cr, is obtained by using the projection operator technique (POT), developed by Argyres and Sigel. The ESRLS function is calculated to be axially symmetric about the c - axis and analyzed by using the spin Hamiltonian $H_{SP}={mu}_B(B{cdot}{^leftrightarrow_{g}}{cdot}S)+S{cdot}{^leftrightarrow_{D}}{cdot}S$ with the parameters g = 1.972 and D = $0.395;cm^{-1}$. In the ca plane, the linewidths show a strong angular dependence, whereas in the ab plane, they are independent of the angle. This result implies that the resonance center has an axial symmetry along the c - axis. Further, from the temperature dependence of the linewidths that is shown, it can be seen that the linewidths increase as the temperature increases, at a frequency of v = 9.27GHz. This result implies that the scattering effect increases with increasing temperature. Thus, the POT is considered to be more convenient to explain the scattering mechanism as in the case of other optical resonant systems.