Arch length considerations due to the curve of Spee: a mathematical model

Arch length analysis should consider discrepancies not only within the sagittal plane but also within the vertical and transverse planes. The vertical deviation of the occlusal plane from a flat plane is known as the curve of Spee. The purpose of this study was to produce a mathematical model of the mandibular arch form in three planes of space and to determine the effect that the curve of Spee has on arch circumference. Two mandibular arch forms, the catenary and the Bonwill-Hawley, were examined. The curve of Spee was modeled as a cylinder perpendicular to the midsagittal plane centered on the arch anteroposteriorly. A mathematical distance formula was used to calculate arch circumferences from the central fossa of the first molars for 10 arches with curves of Spee ranging from 0 to 10 mm. This procedure was repeated for arch circumferences extending from the central fossa of the second molars. Plots for the difference in arch circumferences verses depth of the curve of Spee showed that the relationship between these two variables is not linear and is less than one to one. This model showed that clinical practice of allowing 1 mm of arch circumference for leveling each millimeter of curve of Spee overestimates the amount of arch circumference needed to flatten the curve of Spee.