Dental arch shape: A statistical analysis using conic sections *,* #“ ,A Dr. Sampson Paul D. Sampson, Ph.D. Chicago, Ill. This report addresses two problems in the study of the shape of human dental arches: (I) the description qf arch shape by mathematical functions and (2) the description of variation among the dental arch shapes in a population. A new algorithm for fitting conic sections is used to model the maxillary dental arches of a sample of sixty-six subjects. A statistical model for shapes represented by arcs of conic sections is demonstrated on the sample of sixt?;-six dental arches. It permits the definition of an “average shape” and the graphic representation of variation in shape. The model and methods of analysis presented should help dental scientists to better dejine and quantrfi “normal” or “ideal” shapes and “normal ranges of variation” ,for the shape of the dental arch. Key words: Dental arch, shape, conic sections, average shape, variation in shape T he size and shape of the human dental arch have been studied for over a century. ‘7 2 However, the studies have been handicapped by the lack of good geometric methods for describing or modeling biologic shape and by the lack of a statistical model permitting the investigation of concepts of “average shape” and variation among shapes in a population. Recent work in the field of morphometrics-the measurement of shape, its variation and change-provides valuable new tools for the discussion of biologic shape. 3P6 In particular, Bookstein’s” algorithm for fitting conic sections to data, or Sampson’s’ modification of that algorithm, is well-suited to modeling the shape of the dental arch. Furthermore, Sampson8 provides a statistical framework for analyzing popu- lations of shapes that can be modeled with arcs of conic sections. First discussed in this report are issues relevant to the modeling of shape and to the use of conic sections as models for shape. Included are comments on geometric and statistical aspects of previous approaches to the description of the shape of the dental arch. A brief introduction to the statistical model for populations of conic arcs is presented, and the application of the model is demonstrated with a sample of data on dental arches compiled by the Center for Human Growth and Development (CHGD) at the University of Michi- gan.s A full appreciation of the mathematical details of the statistical model is not neces- sary for an understanding of the discussion of the application. Technical Report No. 109 from the Department of Statistics, The University of Chicago. Support for this research was provided in part by National Science Foundation Grants SOC76-80389 and MCS76-81435. 0002.9416/81/050535+14$01.40/0 0 1981 The C. V. Mosby Co. 535