Rigid-body kinematics and single-tooth displacements

Abstract

T h e branch of physics known as mechanics has influenced analyses and designs of orthodontic appliances and has contributed to the prediction of tooth movement. The form of the orthodontic displacement of the tooth is related to the characteristics of the mechanical action delivered the appliance; the biomechanical relationship, although not fully understood, is a source of sustained'interest in the mechanics of physics applied to orthodontics. The focus of this article excludes the forcedisplacement relationship; rather, the emphasis is on a response to the mechanical action: the displacement and its description. The branch of mechanics (of physics) that pertains to the movements of particles or bodies, without references to causes, is known as kinematics.1 Orthodontists were describing the movements of teeth by their appliances before they realized that mechanics, the applied science, could offer a means of advancing knowledge within the specialty. As the understanding of the mechanics of appliances and the relationships to tooth movement developed within the discipline, the language of mechanics became intertwined with orthodontic jargon in the descriptions of displacements. This mixing of terminology unfortunately has led to incompletely descriptive, often vague, and sometimes mechanically incorrect usage of words and phrases. In submitting this manuscript to the Journal, the need for a review of kinematics applied to orthodontics, specifically, with reference to the description of the displacement of a single tooth is suggested. To begin a review of single-tooth kinematics, several concepts and reasonable assumptions are presented. First, the individual tooth is essentially nondeformable; crown and root(s) cannot move independently. The distance, then, between any two specific points of the tooth is a constant, independent of time, tooth position, or orientation. Second, a reference frame is necessary from which to describe any displacement. The appropriate reference is the familiar, local framework for a single tooth consisting of three mutually perpendicular and intersecting axes; faciolingual, mesiodistal, and occlusoapical describe their directions. The location and angulation of the frame should define the ideal position and