## Introduction

The prediction of arch perimeter gained when expanding the arch or proclining the anterior incisors is particularly important in the decision of tooth extraction vs nonextraction therapy for orthodontic treatment.

## Methods

Correlation of the measured perimeter of the maxillary dental arch and the calculated perimeter applying Ramanujan’s equation for the perimeter of an ellipse was evaluated with 30 diagnostic casts of untreated maxillary dental arches. Both linear and circumferential measurements were made directly on the midbuccal surface of these maxillary arches.

## Results

Ramanujan’s equation had a high level of correlation when comparing the measured perimeter of the maxillary dental arch and the calculated perimeter with the 0.01 level (1.2% error) using a 2-tailed *t *test. The results were applied to Ramanujan’s equation to predict maxillary arch perimeter gained by expansion or proclination of the incisors.

## Conclusions

The ellipse is an accurate geometric model of the maxillary arch form. The average amounts of maxillary arch perimeter gained were 0.73 mm per millimeter of intermolar expansion and 1.66 mm per millimeter of incisor protrusion.

## Highlights

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The perimeter of a maxillary arch was calculated by using Ramanujan’s equation of an ellipse.

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The calculated perimeter was correlated with the measured maxillary arch perimeter (1.2% error).

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We used the ellipse as a model geometric form of the maxillary arch.

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We predicted a gain of 0.73 mm of maxillary arch perimeter per millimeter of first molar expansion.

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We predicted a gain of 1.66 mm of maxillary arch perimeter per millimeter of incisor protrusion.

Arch perimeter or circumference prediction is an essential component when planning comprehensive orthodontic treatment. This is most evident when seeking to resolve dental crowding or arch-length discrepancy (ALD). To treat the ALD, an orthodontist may choose to expand the arch, procline the anterior incisors, distalize the posterior dentition, reduce teeth interproximally, or extract teeth. The decision of the treatment modality must take into account the occlusion, facial esthetics, and stability of the result. Expansion of the arch and proclining of the anterior incisors can be an effective way of gaining arch perimeter when such treatment is indicated. However, without accurate prediction of the arch perimeter to be gained when proclining or expanding the arch in a borderline extraction patient, the orthodontist may underestimate the space gained and erroneously prescribe extractions that will then require excess space closure or, conversely, expect to gain more space and be unable to resolve the ALD.

Ricketts et al suggested guidelines that state that each millimeter of canine expansion provides for a 1-mm increase in arch perimeter, and that 1 mm of molar expansion increased the perimeter by 0.25 mm. Adkins et al estimated the arch perimeter gained in patients treated with a hyrax expander using photographs and pretreatment and posttreatment study casts. Regression analysis indicated that changes in premolar width were predictive of changes in arch perimeter (r ^{2} = 0.69) at approximately 0.7 times the premolar expansion. Germane et al developed a mathematic model using a spline function to compare quantitatively the effects of various types of orthodontic expansion on the mandibular arch perimeter with average arch dimension, intermolar width, intercanine width, and midline arch length. The results of that study indicated that when the canine width and incisor positions were held constant, an initial 1-mm increase in molar width produced approximately a 0.27-mm increase in perimeter, the second millimeter produced an increase of 0.31 mm, and the fifth millimeter of molar width increase was related to a perimeter increase of 0.41 mm. When the incisor positions were fixed, each millimeter of canine expansion provided a 0.73-mm increase in arch perimeter. When arch perimeter was increased by incisor advancement, it was nearly 4 times as effective in increasing arch perimeter as was molar expansion, depending on arch constriction. Hnat et al produced a method for forecasting alterations in arch perimeter related to various width increases in each dental arch based on combined beta and hyperbolic cosine functions using average values for the initial preexpanded arch widths and depths of untreated patients. Because the canine-to-molar width expansion ratios are a function of the point of application of the expansion force relative to the center of resistance of the dentomaxillary complex, 3 canine:molar ratios—1:1, 1.25:1, and 1.5:1—were examined. Hnat et al found that if the maxillary molar width is expanded by 6 mm (3 mm per side) and the canine:molar expansion ratio is 1.25:1, then the arch perimeter alteration is +5.4 mm. Correspondingly, the mandibular arch perimeter alteration is +5.6 mm, when the 2 arches are in an Angle Class I occlusion.

The studies mentioned above represent a range of values that can be difficult to interpret when attempting to predict arch perimeter gained by proclining anterior incisors or expanding the posterior dentition. To accurately predict the arch perimeter, the resulting arch form must be understood. Numerous studies have reported that “normal” dental arches approximate certain geometric curves. Nevertheless, throughout years of investigation, various geometric forms have continually been used to describe the shape of human dental arches, including ellipse, parabola, hyperbola, and catenary curve. Currier concluded that the ellipse is the best geometric figure for describing the form of both the maxillary and mandibular dental arches when landmarks on the facial (outer) surfaces of the teeth are used for comparisons. However, most recently, the arch form has been mathematically modeled more complexly using a computer-aided spline and beta function, bringing into question the accuracy of the classic elliptical arch form.

An ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. The calculation of the perimeter of an ellipse requires an “infinite series” of calculations to be exact. Several mathematicians have attempted the formulation of an adequate approximation. However, the equation formulated by Srinivasan Ramanujan in 1914 is widely considered to be the most accurate. The approximation requires 2 values: *a* and *b* . The semimajor axis (denoted by *a* in Fig 1 ) is half of the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis (denoted by *b* in Fig 1 ) is half of the minor axis.

Perimeter of an ellipse = π ( a + b ) { 1 + ( 3 h / ( 10 − ( 4 − 3 h ) ) ) }