On the cephalometrics of skeletal change

This essay introduces the general tensor analysis of skeletal change for landmark data. Consider first a single triangle of landmarks at two times. Joint changes in the lengths of its sides, or in the positions of its vertices according to some coordinate system, may be taken to specify a uniform deformation of the entire interior. The biorthogonal method expresses this by a pair of principal dilatations--maximum and minimum rates of change in length--along directions lying at 90 degrees in some orientation upon the triangle. No analysis of static form is involved in their calculation, which measures shape change without measuring shape. From this basic biorthogonal decomposition, we pass by a suitable averaging to descriptions of mean change in groups of diverse initial form and subsequently to explicit comparison of two mean changes, such as "treatment effect," all in the same parameters: two dilatations and an orientation. Schemes of more than three landmarks may be analyzed by reduction to triangles. I exemplify the method using data from Sheldon Baumrind's study of Angle Class II treatment effects. With respect to the growth observed in a "control" group of untreated Class II cases, both "cervical" (headgear) and "intraoral" (activator) appliances have the effect of compression a facial polygon horizontally (parallel to S-N) by about 1 percent per year and extending it vertically (perpendicular to S-N) by about 1 percent per day. These effects are slightly larger for the cervical treatment, which also causes an increase in the distance from nasion to the line sella-ANS (that is, "rotates the face downward") by some 1 percent per year relative to the growth observed in the controls.