Euler-angle norms for tooth rotation, torque and tip.

Intro: The aim of the current study was to develop and describe a new measuring system for the orientation of a tooth in a digitalized cast of a jaw and provide new angular values for the rotation, torque and tip of maxillary and mandibular teeth.

Methods: This retrospective cross-sectional study involved the utilization of a sub-group of extrinsic Euler-angles to derive optimal norm values per tooth in three different planes of orientation ('rotation', 'torque' and 'tip') by evaluating the digital representations of the teeth derived from a database containing over 17,500 patients. The process involved the entry of the .stl files of the jaw pairs into a fully automated software system (Smyl:Ai, Ulm, Germany) whereupon jaw alignment, teeth segmentation, landmark identification and visual validation of input files was conducted prior to calculation of the norm values for the three different planes of orientation.

Results: The digital scans in stereolithography (.STL)-file format of the upper and lower dentitions of 1914 individuals with optimal occlusion were chosen and evaluated. New mean (standard deviation) angular values were determined for the rotation, torque and tip of maxillary and mandibular teeth.

Conclusion: The findings facilitate the reappraisal of rotation, torque and tip values currently acceptable as ideal. They will inform anthropologists and dental researchers about occlusion and alignment in orthodontic and non-orthodontic patients and provide baseline data for future studies. The methodology will also enable the evaluation of large numbers of data in relatively short timeframes.