Easier way to find the sagittal depth: comments Berlyn Brixner University of California, Los Alamos National Laborato- ry, Los Alamos, New Mexico 87545. Received 2 November 1981. A recent Letter by Foote 1 presented formulas for calculating the exact and approximate sagittal depth of a chord. Although the formula is correct, the accuracy claimed for the approximate formula (one part in 2000 for d = R/4) is not obtained. We present here a single, simple, and easily remembered formula for calculating both the approximate and, by iteration, the exact sagittal depth. where r is one-half of the chord length, and R is the circular radius of curvature. When the s in the denominator is set at zero, one obtains an approximate formula equivalent to that given by Foote. Table I. Calculation of Sagittal Depths for R = 100 Table I gives the successive approximations of the sagittal depth for several chord lengths in a circle with R = 100. The first entry is the depth given by the approximate formula (s in the denominator set at zero). In most cases, the calculation accuracy is seen to progress at about one significant figure per iteration. The increasingly accurate sequence of sagittal approximations makes it easy to detect errors in the arithmetic. Note that the approximate formula makes an error of about one part in 400 when the chord length is R/5, much greater than claimed by Foote. This work was performed under the auspices of the U.S. Department of Energy, under contract W7405-ENG-36. Reference 1. V. S. Foote, Appl. Opt. 20, 2605 (1981). 976 APPLIED OPTICS / Vol. 21, No. 6 / 1 5 March 1982

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Easier way to find the sagittal depth: comments

Berlyn Brixner University of California, Los Alamos National Laboratory, Los Alamos, New Mexico 87545. Received 2 November 1981.

A recent Letter by Foote1 presented formulas for calculating the exact and approximate sagittal depth of a chord. Although the formula is correct, the accuracy claimed for the approximate formula (one part in 2000 for d = R/4) is not obtained. We present here a single, simple, and easily remembered formula for calculating both the approximate and, by iteration, the exact sagittal depth.

where r is one-half of the chord length, and R is the circular radius of curvature. When the s in the denominator is set at zero, one obtains an approximate formula equivalent to that given by Foote.

Table I. Calculation of Sagittal Depths for R = 100

Table I gives the successive approximations of the sagittal depth for several chord lengths in a circle with R = 100. The first entry is the depth given by the approximate formula (s in the denominator set at zero). In most cases, the calculation accuracy is seen to progress at about one significant figure per iteration. The increasingly accurate sequence of sagittal approximations makes it easy to detect errors in the arithmetic. Note that the approximate formula makes an error of about one part in 400 when the chord length is R/5, much greater than claimed by Foote.

This work was performed under the auspices of the U.S. Department of Energy, under contract W7405-ENG-36.

Reference 1. V. S. Foote, Appl. Opt. 20, 2605 (1981).

976 APPLIED OPTICS / Vol. 2 1 , No. 6 / 1 5 March 1982