## 1 Introduction  
Facial asymmetry is one of the main reasons for patients to undergo orthognathic surgery [1], especially among Asians, where asymmetry between the right and left hemi-mandibles with mandibular pragnathism is common. Currently, bimaxillary surgery with mandibular setback procedures, such as the sagittal split ramus osteotomy (SSRO) or intraoral vertical ramus osteotomy (IVRO), is the main approach to correct this asymmetry. Traditionally, SSRO has been preferred over IVRO in asymmetric surgery due to rapid postoperative recovery and reliable stability through rigid fixation. However, if a large amount of mandibular setback is required for the improvement of facial asymmetry in Asians, IVRO has the advantage of reducing the possibility of injury to the inferior alveolar nerve and less adversely affecting temporomandibular disorders [2; 3]. Recent advances in the field of dentistry allow clinicians to harness the power of 3D imaging, using cone-beam computed tomography (CBCT) among other imaging modalities, to precisely diagnose the patient's skeletal deformity, set up a pre- and post-surgical orthodontic treatment plan, and simulate orthognathic surgery [4; 5; 6; 7]. Nevertheless, to date, when planning bimaxillary surgery for a patient with mandibular asymmetry using 3D CBCT images, the goal is merely to position a menton (Me) located below the mandibular symphysis on the facial midsagittal plane (MSP) as the lateral deviation of the chin is the most characteristic feature of facial asymmetry [8; 9]. However, Me can only be defined as a single point during the two-dimensional (2D) lateral and posteroanterior cephalometric analyses. If the only Me is overlapped with MSP as a treatment objective to correct mandibular asymmetry, then the morphological asymmetry between the two hemi-mandibles likely remain after surgery, especially when asymmetry exists primarily in the mandibular body. In other words, when only considering one Me for predicting surgical results, it may not be possible to capture the exact shape of the mandible and to evaluate stereoscopic asymmetry in the 3D space. To evaluate mandible symmetry in the 3D space, a plane that allows superimposition of the left and right hemi-mandibles is required for using a mirroring technique [10; 11]. With the availability of such a plane, the goal of mandibular surgery for correction of a patient's facial asymmetry would thus be to match it with the MSP to the greatest extent possible by moving the anterior mandibular segment in front of the osteotomy line during the IVRO or SERO procedure. Many studies have aimed to establish this setting plane as a midsagittal plane of the mandible. For example, [PERSON] et al. [12; 13] proposed a midsagittal plane of the mandible comprising of the line connecting the Me, B point, and the middle point between genital tubercles (G). [PERSON] et al. [14; 15] evaluated the structural symmetry of the mandible by using an optimal symmetry plane created by computing the best pairing of the bony voxels on the two hemi-mandibles. [PERSON] et al. [16] measured the residual mandible asymmetry following the menton point correction to the facial sagittal plane by mirroring one side of the hemi-mandible onto the other, and concluded that the symphysis and para-symphysis, and the body and angle of the hemi-mandible exhibited transversal and vertical discrepancies that contributed to mandible asymmetry. However, these studies do not intuitively show the difference in the 3D structure of the two hemi-mandibles by comparing them mainly using two-dimensional (2D) measurement methods such as distance and angle. In our previous study [17], the similarity index (SI), which indicates the ratio of the overlapping volume to the whole mandibular volume when the two hemi-mandibles are superimposed using the mirrored images created by the setting plane, allowed the intuitive assessment of 3D mandibular asymmetry.