Cutting-edge technology is revolutionizing surgery with virtual simulation systems that slash risks, reduce training expenses, and elevate surgical proficiency. These systems hinge on pinpointing precise 3D instrument movements, crucial in minimally invasive surgery. Instead of traditional markers, which slow procedures and pose infection risks, researchers are harnessing raw surgical videos and innovative techniques like background subtraction and deep learning. This groundbreaking approach dissects the 3D tracking challenge into 2D tracking, and then transforms it into 3D motion via geometric dynamics. It ensures instruments avoid critical tissue collisions and blend kinematics, interval arithmetic, and geometric analysis for accuracy. This innovation promises real-time training and autonomous instrument tracking, a game-changer for surgery