Fig. 6.

The pitch, roll, and yaw of a chin segment were measured by comparing the orientation of the reference frame for the chin segment (colored arrows) to the reference frame for the composite skull model (black lines). The red arrow is the transverse (x) axis, the blue arrow is the anteroposterior (y) axis, and the green arrow is the vertical (z) axis. The x, y, and z axes of the reference frame for the planned chin are parallel to the X, Y, and Z axes in the reference frame for the composite skull model. The origin of the reference frame for the chin segment is point 2. The deviation is intentionally enlarged to clearly indicate the method of angular difference calculation used in this study. (A) The first step is to measure the yaw, which is the angle between the x-axis of the reference frame for the chin and the X-axis of the reference frame for the composite skull model as they project on the axial plane (XOY plane, submental vertex view). (B) After the yaw has been calculated, the algorithm rotates the reference frame of the chin around the origin, aligning its x-axis to the X-axis of the reference frame of the skull (removing the yaw, submental vertex view). (C) This image illustrates the frontal view of the chin segment at the same orientation as in image B. After the yaw has been removed, the next step is to measure the roll, which is the angle between the z-axis of the chin and the Z-axis of the skull as they project on the coronal plane (Z–X plane [Au?4], frontal view). (D) The algorithm then rotates the reference frame around the origin to align the z-axis of the reference frame of the chin to the Z-axis of the reference frame of the skull (removing the roll, frontal view). (E) and (F) The last step is to measure the pitch, which is the angle between the y-axis of the reference frame of the chin and the Y-axis of the reference of the skull (left view).