Centers of rotation for combined vertical and transverse tooth movements

For purely transverse orthodontic tooth movements, the center of rotation is defined as that point on the long axis or its extension which remains stationary during the movement and around which the rotational component of the tooth displacement takes place. For tooth movements having both vertical and transverse components, no point on the long-axis line remains fixed in space. The two-dimensional theory proposed herein suggests the more general definition of the center of rotation as that point on the long-axis line which displaced the shortest distance during the tooth movement. The center of rotation can be located for the combined transverse and vertical tooth displacement. It is found to move along a path coincident with a segment of a line in a position depicting the tooth angulation midway through the movement. Formulas, which can be used in conjunction with a composite pre- and post-displacement cephalometric tracing, are presented herein to define the center-of-rotation location for such tooth movements.