A probability distribution of shape for the dental maxillary arch using digital images

Selected landmarks from each of 47 maxillary dental casts were used to define a Cartesian-coordinate system from which the positions of selected teeth were determined on standardized digital images. The position of the i-th tooth was defined by a line of length (l_i) joining the tooth to the origin, and the angle (θ_i) of this line to the horizontal Cartesian axis. Four teeth, the central incisor, lateral incisor, canine and first molar were selected and their position were collectively used to represent the shape of the dental arch. A pilot study using clustering and principal component analysis strongly suggest the existence of 3 groups of arch shape. In this study, the homogeneity of the 3 groups was further investigated and confirmed by the Dunn and Davies-Bouldein validity indices. This is followed by an investigation of the probability distribution of these 3 groups. The main result of this study suggests 3 groups of multivariate (MV) normal distribution. The MV normal probability distribution of these groups may be used in further studies to investigate the issues of variation of arch shape, which is fundamental to the practice of prosthodontics and orthodontics.