3D surfaces based on the Archimedes' spiral are widely used in mechanical engineering, especially in the manufacturing the gear milling cutters. The machining process to create this type of surface is usually performed in specialized machines. In this study, relieving grinding process was conducted to machine the Archimedes surface of HSS P18 workpiece material using a specialized machine (1Б811) to investigate the influence of the grinding wheel graininess (G), hardness of grinding wheel (Hd), grinding wheel velocity (V), and feedrate (s) on surface quality including surface roughness (Ra) and surface hardening (ΔHRC). The experimental plan with 27 experiments (L27) was designed using the Taguchi method. Applying the analysis of variance (ANOVA), the influence of G, V, s, and Hd on Ra and ΔHRC was investigated. G, V, s, and Hd have different influences on Ra and ΔHRC in the relieving grinding process. V has the greatest influence on Ra and ΔHRC with 40.96% for Ra and 26.43% for ΔHRC. The grinding wheel hardness that has the lowest influence on Ra (0.82%), whereas the feedrate has a negligible effect on ΔHRC (7.01%). The interaction effect of the input parameters on Ra is not significant (2.91%). However, for ΔHRC, the interaction influence degree of these parameters on this criterion is quite high (33.77%). From the experimental data, Ra and ΔHRC were presented as a quadratic function of G, V, s, and Hd with high values of determination coefficient (R2 = 92.75% for Ra and R2 = 92.00% for ΔHRC). In the surveyed range of the input parameters, Ra decreases if G and V increase, and s decreases. Also in this survey range, the hardening increases if the values of the input parameters (G, V, and s) increase. This study also determined that there is no clear rule about the relationship between Ra and ΔHRC. Further studies need to be carried out to find the input optimal values ensuring the quality criteria of the relieving grinding process.