Prescription Lenses

661

Prescription LensesWhitney Harris

Determining the prescription of a pairof eyeglass lenses by use of measuredcurvatures uses geometry, algebra, andsome physics. It has an immediacy stem-ming from "hands on" measurementand it has personal relevance for stu-dents who wear glasses. The problem isas follows: given a pair of glasses bor-rowed from a student or an old pairfrom another source, maybe your own,find the type of lens, the base curve usedfor the front surface of the lens, and thepower of the lens. All of this is to bedone with tools and concepts available

to mathematics or science students, and without laboratory equipment.You will need the lens maker’s formula2

1 1 1-=(n-l)(��+��)

RI Ri

where f is the focal length of the lens in question and n is the index of refractionof the lens material (1.52 for glass and 1.50 for acrylic plastic). Ri and Rz are theradii of curvature of the spherical rear and front surfaces of the lens respectively(see Fig. 3). Some conventions are necessary. For a convex surface (bulging out-ward) the radius is considered positive, and for a concave surface (scooped out)the radius is considered negative. Therefore you can expect the focal length f tobear a sign also: positive for a converging type lens (for farsightedness) andnegative for a diverging type lens (for nearsightedness).

Clearly, we have to find Ri and R2. Students thoroughly enjoy using a contourgage to copy the curvature of each surface of the lens and to transfer it to paper.The contour gage is an inexpensive tool available in hardware stores; your schoolshop may even have one you can borrow. It consists of a set of needle-like rodsheld in a frame. When the ends of the rods of the gage are pressed against a sur-face, the rods slide relative to each other to copy the shape. A student who hasthus copied the shape of the lens, will then lay the gage on a sheet of paper andrun a fine pencil along the edge of the gage to duplicate the circular arc formed.With the curve now on paper, the student can use the standard compass con-struction to find the center of the circular arc, and hence its radius. Care must betaken because the arc is small.

School Science and MathematicsVolume 86 (8) December 1986

662Lenses

FIG. 1. Contour gage copying circular arc AC.

There is also an analytic method based on the "sagittal theorem" from geom-etry: If two chords of a circle intersect, then the product of the segments of onechord is equal to the product of the segments of the other chord. Applying this tofigure 2, in which arc ABC represents the lens curve, yields (2R - h) h = a2.Solving for R, and introducing the more easily measured full chord length L =

2a, we obtain

L2 h(1) ^^h-’-T-The small distance h is measured perpendicularly from the midpoint of the chordto the arc. It is a critical value due to being small and appearing in the denomi-nator of the first term. It may help to draw the arc on graph paper with the chordlying on one of the grid lines.


Lenses 663

FIG. 2. Arc ABC is a section of a surface of a lens.

An alternative derivation of equation (1) follows from applying the Pytha-gorean Theorem to the right triangle in figure 2, then solving for R. Incidentally,equation (1) and its surrounding relations are also used to derive the lens maker’sformula.2 By necessity or preference, students can use equation (1) to find Ri andRZ. The only tool needed, besides the contour gage, is a short ruler calibrated inmillimeters.

There follows the measured data and resulting radii calculated from equation(1) for one of my eyeglass lenses (see Fig. 3). For the front (convex) surface ofthe lens:


664Lenses

FIG. 3. Cross section of a lens/or nearsightedness.

h = 2.1 mm., L = 37 mm., Rs = 82.5 mm. = .0825 m.For the rear (concave) surface of the lens:

h = 5.3 mm., L = 44 mm., Ri = -48.3 mm. = - .0483 m.To get a byproduct from substitution of the above data into the lens maker’s

formula, first rewrite it thus

1 1=(n-l)

Ri+(n- 1)-

R?

The first term on the right gives the power of the rear surface of the lens and thesecond term gives the power of the front surface, which optometrists call the"base curve." Actually, in the jargon of optometry, the term "base curve" has adouble meaning. It can be used for the converging power of the front surface ofthe lens or for the front surface itself. The front surface�or base curve�beingmoderately convex, has a pleasant appearance, and reduces lens aberrations aswell. It is chosen from a set of stock values, whereas, the rear surface is customground to suit the individual’s prescription. Substituting the radii values foundabove and the refraction index n = 1.52 for glass yields

= -10.77/m. + 6.30/m.

= -4.47/m. = -4.47 diopters

The last value obtained is the power of the lens in diopters, defined to be thereciprocal of the focal length f, when f is expressed in meters. The negative signtells us that this is a diverging lens. Also notice that the base curve value is 6.30diopters.


Lenses 665

The results compare favorably with my "average spherical" prescription. Theactual prescription of the lens states a base curve of + 6.50 and a lens strength indiopters of -4.25 spherical and -.50 cylindrical. The cylindrical part, super-imposed on the spherical part, gives a slightly greater curvature along a certaindirection perpendicular to the axis of the lens, and corrects for mild astigmatism.The precision of our methods will probably only pick this up as a contribution tothe average curvature of the rear surface of the lens. For simple near or far-sightedness there will be no cylindrical component.To compare the results of their calculations, students may wish to obtain a

written copy of their prescription from their optometrist. This is a good docu-ment to have in any case.

References

1. Heath, Thomas L. Euclid’s Elements, Vol. 2, Book III, Prop. 35, New York: DoverPublishers, 1956:71

2. Miller, Jr., Franklin. College Physics. Fifth Ed. New York: Harcourt Brace Jovano-vich. Inc.. 1982:599

3. Ogle, Kenneth N. Optics: An Introduction/or Ophthalmologists. Second Ed. Spring-field, Illinois: Charles C. Thomas Pub., 1979

4. Rubin, Leroy. Optometry Handbook. Second, Revised Ed. Reading, Massachu-setts: ButterworthCo.. 1981

Whitney HarrisQueensborough Community College�CUNYBayside, New York 11364-1497

HORIZONTAL EARTHQUAKETESTINGA new earthquake testing laboratory at the University of Michigan is testingbuilding durability in a setting that simulates the actual effects of an earthquakeaccording to James K. Wright, Associate Professor of Engineering at the Uni-versity of Michigan.The 9,100-square-foot lab encompasses two sand tanks for geotechnical engi-

neering tests, a holding tank for hydraulic engineering tests, and a "strong wall-strong floor" for structural engineering tests.The "strong wall," made of reinforced concrete with post tensioning is two

feet thick and 20 feet high, and the "strong floor" is four-and-a-half feet of rein-forced concrete.

DICKENS OF A CHRISTMASChristmas actually begins in Hanover well before Dec. 25 as this Yankee collegetown takes on the aires of merry olde England December 6-7. Carolers andmimes stroll the small town main street Charles Dickens could have known andloved. Christmas goose dinners are served at Dartmouth’s old Hanover Inn bothFriday and Saturday nights. Hanover’s famed stores stay open late, minded byfolks in Victorian costumes. There’ll be fireworks on the college green and aChristmas ball with music by a jazz band in Dartmouth’s famed Hopkins Cen-ter, which is attached to the Hanover Inn.


Prescription Lenses

Documents

Transcript of Prescription Lenses