To analyze hard turning performance characteristics, a new mathematical model was developed for the hard turning process, and cutting force (CF), another important response for cutting machining, was also studied in the present work. The analysis of the mathematical model and experimental results revealed that thrust force (Fy) was the largest, followed by tangential force (Fz) and feed force (Fx). The resultant CF was most influenced by inclination angle (IA) with 25.02%, followed by rake angle (RA) (14.26%) and cutting edge angle (CEA) (10.04%). Increasing CEA changed the position of cutting on the tool-nose radius and increased local negative RA and correspondingly local clearance angle (CA). Meanwhile, increasing negative RA and IA resulted in larger local negative RA and CA. Moreover, local RA and local CA were the main geometric factors affecting surface roughness (SR), tool wear (TW), and CF. Increasing local negative RA resulted in higher SR and CF. In contrast, increasing local CA resulted in lower SR, TW, and CF. Under specific conditions, the positive effects of the local CA overcame the negative effects of the local negative RA, leading to a simultaneous decrease in SR and TW. The proposed novel mathematical model can be further applied to calculate local CF, cutting temperature, and TW for each cutting-edge element, to analyze and optimize the hard turning process.