Instantaneous center of rotation, the first step to build up the digital laboratory of complex motions.

Calculating instantaneous centers of rotation to describe combined rotational and translational motions has a long history in many fields of applied science and basic rigid body kinematics. However, only some theoretical studies have explored the fundamental characteristics of this system. This study used digital three-dimensional modeling and computing methods to examine the system's operation in a controlled in vitro-like environment. The effects of inaccurate registrations on the resulting motion were also analyzed. We registered 28.65, 14.33 and 9.55 EcD_ratios for 2°, 4° and 6° of closure respectively, and described a structured, predictable framework based on a solid mathematical background. Our findings align with previous publications, indicating that the longstanding debate over the pure rotation of the temporomandibular joint arises from misinterpretations of scientific findings due to a lack of fundamental knowledge of the basic characteristics of the system. Our simplified geometrical approach significantly reduces the complexity of the existing complex kinematic model, making it more accessible for practical applications, easier to understand, and potentially more applicable in orthopedics or temporomandibular joint radiology. We identified five fundamental characteristics of the system as we described the effects of the acting translational component in the complex motion. We also presented a detailed model concerning the effects of inaccurate rotation axis registration on the resulting compromised transformation, improving our understanding of the error tolerance level of articulation systems. Our results show that the system might tolerate errors as great as 3-4 cm in some settings in the parallel error direction, while in case of circular and perpendicular error types an approximately 2 mm axis registration error would exceed the clinically desirable 0.1 mm occlusal error level. Our experimental modeling strategy might provide extensive data for machine learning and for analyzing and comprehending the fundamental characteristics of different complex motion systems in the future.