Thank you for spotting an error in the printed formula (Chung DD, Wolfgramm R. Maxillary arch perimeter prediction using Ramanujan’s equation for the ellipse. Am J Orthod Dentofacial Orthop 2015;147;235-41). In the denominator, as you pointed out, it should have been printed 10 + 4 − 3 h , instead of 10 − 4 − 3 h . We used the correct formula for all the calculations in the manuscript, and the printed error does not change the reported results.
You mentioned that Germane et al (1991) and Braun et al (1998) proved arch forms with spline and beta functions, respectively. Arch forms can also be modeled precisely with other curve-fitting methods, such as high-degree polynomial equations. Without specifying all the coefficients generated by the procedure of fitting a curve, it is meaningless to say that a dental arch is a cubic spline or a fourth-degree polynomial. To calculate the perimeter of the dental arch by using a spline function, the length of a segment between all data points must be added numerically; it is tedious work requiring significant computational time without a computer. Although a beta function can generate a coordinate of arch depth (y) at width (x), it cannot explicitly obtain a perimeter value. We verified that the shape of a maxillary arch was very close to an ellipse by comparing the measured perimeter and the calculated perimeter using Ramanujan’s equation. The beauty of this equation is that the perimeter can be easily obtained as a function of only 2 variables, semimajor axis a and semiminor axis b . Since the transverse distance between the molars and the arch length was simulated by the length of the semiminor and semimajor axes, any change of the perimeter by altering values a and b can be easily obtained with a hand-held calculator.
Your letter suggested that Ramanujan’s equation should be used only for ideal or theoretical dental arches, which demonstrate symmetry. Although we encounter few patients with severe unilateral constrictions or skewed dental arches, for those atypical ones, we still find that predicted perimeter is useful. In the rare constriction case, a predicted perimeter can be obtained by using the semiminor axis of the unconstricted side. This predicted perimeter can be compared with the existing perimeter to determine whether unilateral expansion of the constricted side will resolve the crowding. In subdivision patients, when the first molars are not in symmetrical positions anteroposteriorly, the predicted perimeter is also useful. In these patients, a transverse axis can be drawn from the molar with the correct anteroposterior position to its contralateral midbuccal point. By altering the value of a or b , a clinician can predict the calculated perimeter and determine whether the measured crowding between those 2 points can be resolved.
In the discussion section of our article, we stated that the use of second molars does generate a better fit of an ellipse to the dental arch. However, we selected the first molars because the second molars are not always erupted in many patients at the time of diagnosis. Additionally, most expansion appliances use the first molars for banding. To make a better theoretical fit, we used the midbuccal point of the distobuccal cusp, as opposed to the mesiobuccal cusp.
You suggested that Ramanujan’s equation should be used only for oval or elliptical arches, not V- or U-shaped arches. Contrary to common belief, an ellipse is not a single form, but it can take many forms. Eccentricity (e), the important parameter of a conic section, can be used to describe an ellipse. The ellipse has 0<e<1, a circle has e = 0, and a parabola has e = 1. Therefore, a wide variety of shapes (even close to a circle or parabola!) can be approximated by an elliptical equation.
Again, thank you for finding our printed error and allowing us to clarify some of your concerns.