## 1 Introduction  
The spatial orientation of the three semicircular canals (SCs) has been a matter of interest since early anatomical studies of the inner ear and temporal bone. Recent clinical studies show that SCC spatial orientation plays a main role in specific pathologies (i.e., positional vertigoes [1]). In addition, the specific morphology of the vestibular system can explain the variability on the induced adverse symptoms (i.e., vertigoes and dizziness) on patients that are subjected to MRI high magnetic fields [2; 3; 4]. [PERSON] first, and then [PERSON], [PERSON] and [PERSON] later, in the transition from the 19 th to the 20 th centuries, gave the first measurements of the angles between the 3 SCs. The methodology was mainly based on corrosion casts and major differences were seen between the results given by different authors [5; 6]. Since then, other methods have been developed to study the anatomy of the SC post-mortem. With the advent of modern imaging technology and in particular with the ability to assess axial scans and perform reconstructions in several planes, more detailed in vivo evaluations in higher amounts of bones have been performed. However, even for computed-tomography (CT) and magnetic resonance imaging (MRI) studies, the differences in results are still significant, although of lower magnitude than before. CT scans [7; 8] show some variability in results except for the angle between the horizontal SC and posterior SC, where results are more consistent. When compared to results in more recent anatomical measurements made in explanted temporal bones [9], the differences are smaller, and can, in general, be attributed to the exclusion of some parts of the canal in the analysis (common crus and ampulle). In the case of MRI, data also differ and the angles between canals (in the same ear) are close to \(90^{\circ}\) in the case of the anterior and posterior SC [10] as well as in the case of the anterior and horizontal SC [11]. In general, the canals do not lie in specific planes in the space. Since the first measurements, it became clear that the canals follow a sinuous course and only their mid-portion is appropriately in a plane; that portion makes an obtuse angle with the ampullated and non-ampullated ends. In addition to this, other attributes of the canal (non-planarity, radius of curvature and length of the common crus), well documented with micro-CT scans [12], render the spatial orientation characterization task complex and elusive. The relevance of spatial orientation to normal range of function dates back to the very first experiments in vestibular physiology. Since Crum-Brown, coplanarity (as documented and experimented in the horizontal canals) was extended to the six SCs. The planes for the left anterior canal with the right posterior (LARP) and for the right anterior with the left posterior (RALP) considered, as anatomical and functional units, became a matter of anatomical interest throughout the years. More recently, since their specific evaluation has become a standard in vestibular testing at bedside [13], the interest has grown. Previous experimental works based on post-mortem dissections [9; 14] or imaging [7; 10; 15] have a common nexus, the data processing protocol and extracted information always depend on subjective factors that are involved on how to delimit the section of the canal that must be considered in the analysis. In all the previous works, this is the decision of an expert operator. Recently published works by [PERSON] et al. proposed a mathematical method that allows 3D measuring and modeling of the semicircular ducts in living humans [12]. Their mathematical method automatically reconstructs the semicircular ducts geometry from computed tomography images validating the results with micro-computed tomography as ground truth. The final result is a centroid path of the cavities that allows extracting global characteristics of the vestibular cavity. Although the method proposed by [PERSON] et al.