Mechanical-Mathematical Modeling of the Shape of Dental Arches for Orthognathic Occlusion

We perform mathematical modeling of the shape of dental arches. It is assumed that dental arches can be modeled by second-order curves. As a result of the analysis of the data of measurements performed for the diagnostic models of dentures, we compute both the coefficients of the quadratic form of the indicated second-order curves and the invariant of this form. Since the invariant of the second-order curve appears to be greater than zero, we make conclusion that the shapes of dental arches are described by the equation of ellipse. The parameters of the canonical equation of ellipse used to model the shape of dental arches are determined. We also develop a procedure for specifying the parameters of the model with an aim to use them in the processes of orthopedic and orthodontic treatments.