Cross- and triple-ratios of human body parts during development

Recently developed landmark-based geometric morphometry has been used to depict the morphological development of organisms. In geometry, four landmarks can be mapped to any other four by Möbius transformations, if the cross-ratio of the landmarks is invariant and vice versa. To geometrically analyze the morphological development of the human body, we examined the cross-ratio of three consecutive body parts that are segmented by four landmarks in their configuration. Moreover, we introduced the triple-ratio of five landmarks that segments four consecutive parts (e.g., the shoulder, upper arm, forearm, and hand) and examined their growth patterns. The cross- and triple-ratios of the upper limb and shoulder girdle in fetuses were constant when biomechanical landmarks were used, although the cross-ratio of the upper limb varied when anatomical landmarks were used. The cross-ratios of the lower limbs, trunk, and pelvic girdles in fetuses differed from their corresponding cross-ratios in adults. These results suggest Möbius growth in the fetal upper limb and shoulder girdle but not in the other body parts examined. However, the growth balance of the three contiguous body parts was represented by the developmental change in the cross-ratio. Therefore, the cross- and triple-ratios may be applicable for simple but significant assessments of growth balance or proportion of the body parts.