The geometry of craniofacial growth invariants

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