The geometry of craniofacial growth invariants

In the description of craniofacial growth it is crucial to identify constant geometric relations. The search leads to complex manipulations of coordinate data from cephalograms. In the regularities that he finds, the clinician must be aware of the difference between empirical contingency and geometric necessity. For instance, common procedures for identifying centers away from which growth proceeds radially (the "polar phenomenon") ignore theorems about the indeterminacy of such centers. The scan for growth invariants in configurations of landmarks should be based on the method of tensor analysis. Any shape change may be summarized in terms of two directions in each triangle of landmarks. The directions are at 90 degrees in each triangle both before growth and after; one of them bears the greatest percent rate of change of length of all directions in the triangle, and the other bears the least. From a diagram of these axes one comes immediately to strict invariants of the shape change--specific angles and proportions along segments defined by relations between the triangle and the axes--linked to other specific angles and proportions which change most rapidly. It is simple to adapt this machinery so that it can deal with average shape changes in populations. Included here are two examples of the technique: (1) extracting the statistically most stable definition of "growth axis" (displacement of menton from cranial base) in children aged 6 to 14 years and (2) describing the shape change of the rat calvarium between the ages of 7 days and 150 days. Such descriptions should be the first step in designing any cephalometric analysis.