None of the other frames of reference is changed by the flexion in the atlanto-occipital joint, therefore it is possible to move on to the lateral rotation. The axis of rotation is through the center of the odontoid process so there is an offset for the atlas and the occiput. The quaternion for a 30° left lateral rotation is given by the following expression.
The center of rotation for the atlas is 0.5 units anteriorly along the sagittal axis of the vertebra. The center of the atlas is elevated 0.25 units above the Q point, which is the origin of the coordinate system for these calculations. The vector that connects the center of rotation for the atlas to the center of the atlas is . The location for the center of the atlas in the coordinate system centered upon the center of rotation after the 30° lateral rotation is computed as follows. However, this vector extends from the center of rotation, but we want it is the universal coordinates, so we must add the value for the center of rotation to obtain the atlas position in the universal coordinates.
The main effect is to shift the center of the axis about a quarter of a unit laterally. For rotations of much more than 45°, there would be a danger of impingement of the spinal cord between the axis and the atlas. This may be why lateral rotation is generally limited to an excursion of about 45°.
The axis of rotation is not changed by rotation about itself. The frame of reference is not affected by the offset, but it is affected by the rotation.
The net effect is to rotate the frame of reference in the horizontal plane.