## 1 Introduction  
Achieving a high level of predictability in an orthodontic treatment has consistently proven to be a motivating and demanding challenge to orthodontists, primarily due to various factors that could influence the clinical outcome. That is why a better understanding of how to achieve tooth movements would allow the design of optimized appliances, leading to more efficient orthodontic treatments. However, in addition to playing an important role in smile aesthetics and functional occlusion, the accurate positioning of the maxillary canines that support the alar base and upper lip should be the goal of orthodontic treatment [1]. Today, thanks to improvements in the biomechanics and physical characteristics of materials [2], aligner therapies have the possibility to treat several case typologies such as Class II, Class III and interdisciplinary cases [3; 4; 5] with an efficacy comparable to that of braces, differently from the early age where clear aligners were only considered to treat simple orthodontic problems like mild or moderate anterior crowding. The use of auxiliaries and attachments with aligners is widely adopted in clinical practice thanks to the good mechanical and chemical properties of the aligners [6; 7], and to the advantages in using removable appliances which require fewer emergency visits and allow better oral hygiene, increasing the possibility to achieve complex movements such as rotation or torque [8]. The digital workflow to design and manufacture orthodontic aligners relies on the digital reconstruction of a patient's anatomy and CAD/CAM (computer-aided design/computer-aided manufacturing) techniques [9; 10]. Specialized dental technicians use CAD systems to segment the digital patient's anatomy, creating separate objects for each tooth and soft tissues. Afterwards, following geometrical and functional constraints, the technicians reposition the teeth to develop a digital treatment plan: the entire treatment is then divided into smaller sequential tooth movements, from the baseline condition to the final expected outcome. For each treatment step, a 3D model of the dental arch is 3D-printed and the customized appliances are then produced from the 3D-printed models by a vacuum thermoforming process. The current workflow is based mainly on the analysis of tooth crown positions, with little consideration for the root movements. This approach could lead to erroneous estimations of the location of the tooth's center of resistance and to dangerous interferences with roots during the treatment, compromising its outcome. The rotation movement of conical-shaped teeth is considered a difficult movement to achieve with clear aligners. According to a prospective study realized on 53 maxillary canines, the mean accuracy of the rotation movement was 36% [11]. For this type of movement, the maxillary canine reports an accuracy lower than other teeth such us the maxillary central incisor (55%) and the mandibular central incisor (52%). The lowest accuracy was noticed for the mandibular canine, reporting an accuracy of 29%. The accuracy of the rotation movement was significantly reduced for rotations greater than 15\({}^{\circ}\) (19%) [12]. Additionally, another study investigating the mesiodistal angulation of canines with fixed orthodontic treatment revealed the importance of movement control of the lower canines, especially in orthodontic mechanics in the mandible, due to the increase in the mesiodistal angulation of the lower canines in all malocclusion groups [13]. In these studies, the accuracy of tooth movements, such as rotation or distalization, was calculated by superimpositions of virtual models by using reference surfaces such as palatal rugae, untreated teeth and dental implants [8; 11; 12]. By comparing pretreatment optical 3D scans with the virtual plan and with the post-treatment optical 3D scan, it was possible to calculate the accuracy of different tooth movements.