A mathematic-geometric model to calculate variation in mandibular arch form.

A mathematic-geometric model was used to evaluate the variation of mandibular dental arch length with respect to the incisor inclination, but without modifying the intercanine width. In analytical terms, the equations of the curves representing the lower dental arch, before and after incisor inclination of 1 mm and of 1 degree, with controlled and uncontrolled tipping, were studied. The length of the mandibular dental arch changed in the parabolic arch form by 1.51 mm for each millimetre of incisor inclination with respect to the occlusal functional plane, by 0.54 mm for each degree of controlled tipping and by 0.43 mm for each degree of uncontrolled tipping. In the elliptical arch form (e = 0.78), it changed by 1.21, 0.43, and 0.34 mm, respectively, in the hyperbolic form by 1.61, 0.57, and 0.46 mm, in the circular form by 1.21, 0.43, and 0.34 mm, and in the catenary form by 2.07, 0.74, and 0.59 mm. The results show that by changing the arch form without modifying the dimension of the dental arch, different arch lengths can be gained for each millimetre of proclination. In addition, by controlled tipping an inter-incisive arch one-fifth longer than by uncontrolled tipping can be obtained. It would be advisable in orthodontic treatment planning to evaluate the type of dental arch, since the space available or the space required changes depending on the arch form and on the orthodontic tooth movement.