## 1 Introduction  
The history of facial deformity correction surgery has evolved through several phases. At first, this surgery was only intended for the correction of the most evident deformities that could not otherwise be corrected by orthodontics alone [1]. In these situations, it is obvious that the aesthetic objective was secondary to the functional one [1]. Subsequently, with the progressive reduction in surgical risk, the treatment's objective shifted progressively from the functional to the aesthetic [2]; this is also due to modern society increasingly imposing a pleasant facial aspect on people that has become more and more important for social acceptance and self-esteem [3, 4]. Therefore, since facial aesthetics has become the main reason for orthognathic surgery, it is evident that the anthropometric examination of the face has become more and more crucial in the planning of the sites and extent of skeletal repositioning [5, 6]. Facial aesthetics, once considered almost only in a profilometric view, is now analysed in the three dimensions of space through increasingly sophisticated methods that allow a very realistic three-dimensional reconstruction of the data and, at the same time, a precise simulation of the surgery and its possible effects on the final harmony of the facial skeleton [7, 8]. Dento-maxillofacial deformities can be classified as sagittal (class II and III), vertical (long and short face), or asymmetric. However, even in cases where the main problem is sagittal or vertical, an asymmetrical component is often associated. This finding suggests that an asymmetric component is usually present [9] and the treatment plan must take this into account to achieve the best result [10]. Correction of these problems is not simple, as it is well known that in asymmetrical deformities the skeletal structures are not normal structures arranged in an asymmetrical position in space [11]. They are inherently asymmetrical structures because they have grown in a distorted muscular environment [11]. This is particularly true for the mandible, which in situations of asymmetry generally has quite different borders and angles on the two sides [12]. Consequently, the correction of a mandibular symmetry cannot be limited to the centring of the chin in relation to the midline of the face but must include a thorough correction of the lateral and angular regions of the face as well, especially in complex cases [13; 14; 15]. Above all, planning the angular region of the face turns out to be very important to obtain a final harmony of the face that is completely satisfactory for the patient, and it must be conducted with appropriate methods. To understand the movements that the mandible will undergo during surgery, we can imagine the dentition of the lower arch as a triangle defined by three points A, B, and C (Figure 1). If the orthodontist has adequately performed the pre-surgical preparation, the dental inter-incisive midline is aligned to the centre of the chin (A point). The two molar regions, right and left, are related to the two angular regions (B and C points). The surgeon can move this triangle in space in various ways. In the sagittal direction, the surgeon can move it forward or backwards to correct a second or third skeletal class malocclusion. The surgeon can also rotate it in various ways to correct an asymmetry. When the centre of rotation shifts near the anterior region, the mandibular angles position will change in the transversal dimension much more than the chin (Figure 2(1), yellow point). In this example, the centre of rotation is located at A point. Alternatively, when the centre of rotation shifts back near the molar region, the chin position (A point) will shift much more in the transversal dimension than the region of the angles (Figure 2(2), yellow point). Figure 1: The mandible seen from above: it may be imagined as a triangle. Figure 2: The centre of rotation (in yellow) substantially changes the geometric behaviour of this triangle.