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$: Shape and Refractive Powers in Corneal Topography · 2017-05-02 · Shape and Refractive Powers 2097 P, = (n- \)/r, {5) FIGURE I. A cross-section (meridional plane) ol the corneal$
Shape and Refractive Powers in Corneal Topography

Stanley A. Klein and Robert B. Mandell

Purpose. To compare the relative advantages and disadvantages of four different representa-tions of corneal power—identified as instantaneous, axial, position, and refractive—basedon curvature, slope, coordinate position, and focal properties, respectively.

Method. The lour types of corneal power were evaluated by examining their interrelationshipfor 12 hypothetical corneal shapes chosen to represent the general characteristics of regular,irregular, and surgically altered corneas.

Results. There is only a limited association between refractive power and the other threepowers, which are based on shape properties. For corneal shapes represented by ellipsoidsof low eccentricity (e), refractive power increased as a function of the distance from thereference axis, whereas axial and instantaneous powers were either constant (for e = 0) ordecreased, with instantaneous power having the largest decrease. Refractive power decreasedfor elliptical corneal shapes with eccentricities greater than the reciprocal of the index ofrefraction, although always decreasing less than the axial or instantaneous power. For variouscorneal shapes formed by segments of circles (or a polynomial) with large differences ininstantaneous power, the refractive power was more closely associated with axial than instanta-neous or position powers.

Conclusions. The Purkinje image measurements from videokeratography are most closely re-lated to the slope-based axial power because it is slope that determines the direction ofreflected rays forming the corneal image. Because of this direct connection, axial power isless sensitive to noise than is refractive, instantaneous, or position power. Present videokerato-graphs that report axial power provide an approximation of refractive power, but if an exactrefractive power is needed it can be calculated easily. Instantaneous power provides the mostsensitive measure of local curvature changes, such as those occurring in keratoconus orrefractive surgery. There are unique practical applications for each of the four powers. InvestOphthalmol Vis Sci. 1995;36:2096-2109.

An the field of cortical topography there is ambiguityfor the term corneal power, which implies refractivepower but. is usually applied to one of various expres-sions of corneal shape.1 Although the use of the term"power" as a shape expression is a misnomer, it hasbecome accepted through popular use, is generallyunderstood, and will be followed here for consistency.

BACKGROUND

The refractive power (sometimes called "local"power, Pf) will be defined as:

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P, = n/f (I)

where n is (he index of refraction of the corneal me-dium and f, the secondary focal length, is the dis-tance from the corneal vertex to the intersection of arefracted incoming parallel ray with the referenceaxis. Throughout this article, we shall consider theeye's optical system as a single refracting corneal sur-face. Iii the paraxial region, the refractive power canbe expressed in terms of the paraxial radius of curva-ture, r.

P = ( n - \ ) / r (2)

where n is typically chosen to be 1.3375 or 1.376, de-pending on whether the entire cornea or only theanterior surface is to be considered.2

It is possible to use equation 2 as the basis for a

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Shape and Refractive Powers 2097

P, = ( n - \)/r, {5)

FIGURE I. A cross-section (meridional plane) ol the cornealshape, together with the parameters that are used to specifythe three shape-based corneal powers at height, y. All calcu-lations are for rays lying in the meridional plane. Axialpower is based on the distance, d,,, from the cornea alongthe corneal normal to the reference axis, and instantaneouspower is based on the distance, r,, to the instantaneous cen-ter of curvature. Position power is based on the distance,df,, from the corneal point to the center of curvature for thevertex. The distance z represents the corneal sagittal depth.

set of specifications of corneal shape. We considerthree options for how to measure the distance r at acorneal point, y, which results in three choices forspecifying shape-based power (see Fig. 1). Positionpower is defined as:

l>t,= ( n - \)/dt) (3)

where d,, is the distance from the cornea at point y tothe paraxial center of curvature. We use the paraxialcenter of curvature instead of alternative referencepoints so that all four definitions of power are identi-cal in the paraxial region. For asphcrical corneas, theline connecting the paraxial center of curvature tocorneal points usually will not be perpendicular to thecornea; it has geometric rather than optical signifi-cance. In equation 43, the position power will be di-rectly related to the fluorescein pattern. Axial poweris defined as:

/>, = (n - 1)M, (4)

where dn is the distance along the corneal normal fromthe corneal point to the reference axis. The three-dimensional normal might not intersect the referenceaxis. However, in the current study, all calculationswere performed meridian by meridian with all quanti-ties constrained to lie in the meridional plane. In thatcase, the corneal normal (in the meridional plane)must intersect the reference axis. Instantaneous poweris defined as:

where r, is the instantaneous radius of curvature inthe meridional plane.

The three shape-based powers are all specified interms of the distance from a corneal point to a refer-ence point. The reference point is specified by twoconditions. For dp, the two conditions are indepen-dent of the corneal point, and both come from thecorneal vertex (the point at which the reference axisintersects the cornea). The vertex normal and vertexcurvature determine the paraxial center of curvatureused for defining dp. For da, one condition comesfrom the corneal normal and the other from the ver-tex normal. The intersection of the normals from thevertex and the corneal point gives the location of thereference point. For r,, the reference point is indepen-dent of the vertex, and both conditions come from thecorneal point (the normal direction and curvature). Itis for this reason that the three shape-based definitionsof power can be called position-based (Pt>), slope-based (/'„), and curvature-based (P,) powers.

Refractive power was defined in equation 1 as Pt

= n/f, where / ' is the distance from the cornealvertex to the point on the axis intercepted by an in-coming parallel ray that has been refracted at the cor-nea. Figure 2 shows the construction that is used forspecifying refractive power. For paraxial rays, the re-fractive power equals the shape-based powers, Pj = /-*,= P,, = Pt>, as long as the same index of refraction isused for both refractive power and shaped-basedpower. For off-axis rays, f is calculated from ray trac-ing by using Snell's Law.

Throughout this article, two simplifications will bemade. First, we assume that the reference axis is nor-mal to the corneal surface (the case of a nonnormalreference axis, such as the line of sight, will be consid-

FIGURE 2. Construction for specifying refractive power. Foran off-axis ray at height y, the focal length f is calculatedfrom ray tracing by using Snell's Law. The distance z repre-sents the corneal sagittal depth.

2098 Investigative Ophthalmology & Visual Science, September 1995, Vol. 36, No. 10

cred elsewhere). Because it is a surface normal, thereference axis will be the conical optic axis for refrac-tive considerations. Second, our derivations will becarried out in the meridional plane, the plane thatcontains the reference axis. When discussing refrac-tive power, we will force the rays to lie in the meridio-nal plane by assuming that the conical shapes arerotarionally symmetric around the reference axis (ro-tate the two-dimensional shape for positive values ofy around the axis). By that assumption, any ray arrivingparallel to the optical axis will stay in the meridionalplane after refraction. However, for the three shape-based definitions of power, it is not at all necessary tobe restricted to surfaces of revolution because the two-dimensional shape in the meridional plane is well de-fined for general shapes. Our analysis is applied sepa-rately in each meridional plane.

Corneal Shape Characteristics

The human cornea has some general shape character-istics that can be identified and that form the basisfor a comparison of the four defined powers. Thesecharacteristics are:

(a) A central radius of curvature between approxi-mately 4 and 9 mm, which includes the conicalabnormalities of keratoconus and postrefractivesurgery.

(b) A periphery with a radius of curvature that in-creases continuously and that for a first-orderapproximation can be simulated by an ellipse,'1 'except, for those corneas included under thecharacteristics c to e.

(c) A periphery' that is irregular by having discreteareas, each with nearly constant curvature.(>/

(d) Postrefractive surgical corneas in which there isa central flattened area superimposed on an oth-erwise normal corneal shape. The transition canbe cither smooth or abrupt."

(e) Keratoconus and other ectatic anomalies inwhich the cornea has a discrete area of highercuivature either centrally or peripherally.'

(f) An anomalous cornea with very small areas thatare so highly irregular (such as scars) that theycannot be measured.

All but characteristic (f) will lend themselves tobe evaluated using 12 hypothetical corneal shapes, forwhich we calculate the four power representations.On this basis, it is possible to compare the relativeadvantages and disadvantages of the four definitionsof corneal power.

Relationships Between the Four CornealPowersKlein and Mandcll1" derived a simple expression forP, as the derivative of yPa with respect to y, the cornealposition:

/J,()') = d\y PH(y)]/Hy = PH + ydPJdy. (6)

The inverse relationship giving Pn in terms of P,is:

\(y) =~ f P,{y')dy'.v J

(7)

This latter relationship is simply stated as: the axialpower at position y is the average of all the instanta-neous powers from the axis to )'.

It will prove useful to calculate the difference be-tween the instantaneous and axial powers. For a sur-face of revolution, the axial power equals the powerin the sagittal plane, 90° from the meridional plane,so this difference is the cylinder power. From equation6 the difference is seen to be:

/>yi,n<.,,(>') = P,(y) - PAy) = ydP,,(y)/dy. (8)

An alternative expression can be found by sub-tracting /J,()') from both sides of equation 7:

v

»(y) - P,(y) = - f t W ) - P,(y)W-V J

(9)

To calculate the conical position needed for posi-tion power, P/n and refractive (focal) power, Pj, wemust first calculate the corneal slope from the axialpower. From Figure 1 and equation 4 it is found that:

sin(0) = y/d,, = yPJ(n- 1). (70)

The sine can be converted to a tangent using thetrigonometric identity:

tan(0) = si (11)

Knowledge of the tangent leads directly to thecorneal position:

(12)z(y) = \ — dy = tan(0)rf)i.

As will be seen, in many cases a simple analyticexpression for conical shape can be obtained, but, ifthat is not possible, the position can be obtained fromequation 12 by numerical integration.

Equation 12 shows how to calculate surface posi-tion from surface slope, defined with respect to they axis. Because information about the slope is onlyavailable at each image ring of the vidcokeratograph,a smoothness assumption must be made for how to


interpolate the slope between rings. The simplest waythis may be accomplished is to use a simple sphericalshape between rings; the problem is that it producesa discontinuous jump in curvature at each ring."""Other shapes can be used to improve the interpolationbetween rings, such as a cubic spline algorithm usedby Klein" for calculating corneal position with contin-uous curvature. This ambiguity in the interpolationfunction is seen as a disadvantage by those who prefera method that measures position directly. However,the ambiguity is expected to be small (position accu-racy from equation 12 is approximately the same asfound by the instruments that measure position di-rectly) as long as the sample points arc close together(approximately 0.5 mm), and an algorithm with con-tinuous curvature is used. The algorithm proposed byKleinM is designed to achieve this interpolation forsurfaces without axial symmetry.

To calculate the position-based power, Pt, usingequation 3, we must first calculate dt>, the distancefrom the corneal point to the paraxial center of curva-ture. From Figure 1, it is seen that dt, is given by thefollowing Pythagorean sum:

d,,= fVn U3)

where ris the paraxial radius of curvature. Thus, Pt, isgiven by:

Pp= (n- (14)

The fourth definition of power is the refractivepower, Pj. It differs from the previous three definitionsin that it based on the refractive properties of thecornea rather than simply on corneal shape. The cal-culation of local power for off-axis corneal points isderived from equation 1 (see Fig. 2):

Pi = = n/\z + ytan(0 - 0,)] (15)

where f is the distance from the corneal vertex to thepoint where the refracted ray intercepts the referenceaxis, 0 is the angle of incidence that was definedthrough equation 10 in terms of P(l, the axial power,and 0, is the angle of refraction given by Snell's law:

sin(0.) = sin(0)/n. (16)

Figure 2 illustrates these various quantities. Theposition z is the same quantity that was defined byequation 12. We have now constructed the stepsneeded to calculate all four of the powers defined.Given any one of the three shape-based powers, wecan calculate the other two by integration or differen-tiation and then can calculate refractive power.

-15

-5 0 5corneal distance (rom axis (mm)

FIGURE 3. Diagram of the relative relationships of cornealshapes 1 to 4, represented by conies of eccentricities 0 (cir-cle), 0.5, \/n, and 1.0 (a parabola). In the central regionrepresented by the cornea, the z positions barely are distin-guishable from each other. The curves are drawn to scale.

METHODS

We compared the four definitions of corneal powerusing 12 hypothetical corneal shapes designed to testthe general properties of normal and irregularlyshaped corneal topography. These shapes are repre-sented diagrammatically in Figures 3 to 6. Shapes 1to 4 (Fig. 3) are ellipses of varying eccentricity de-signed to represent corneal characteristics (a) and (b)(see the list presented earlier in the section entitledCorneal Shape Characteristics) for the normal range.Shapes 5 to 8 consist of two segments of circles (i.e.,segments with constant radius and instantaneouspower), with the central segment having a longer ra-dius. They arc designed to represent conical charac-teristics (c) and (d) for corneas after refractive sur-gery. For shape 5 (Fig. 4), both segments have theircenters of curvature on the optic axis, so that at thepoint where the two segments meet, there is an abruptchange in slope. Shapes 6 and 7 arc the same as shape5 but are deccntcrcd by rotating the cornea to newcentral reference axes. Shape 8 (Fig. 5) has the sametwo radii of curvatures as shapes 5 to 7, but, in thiscase, the slope is continuous where the two circlesmeet so that the center of the outer circular segmentis not on the reference axis. Shapes 9 to 12 representcorneal characteristic (e) for different forms of kerato-conus. Shape 9 (Fig. 6) consists of an inner and anouter circle with centers on the reference axis. Theinner circle has higher power representing the con us.To provide a smooth transition between the two cir-cles, we have added a third circle of larger radius,whose center is not on the optic axis. The slopes arcsmoothly continuous across all the transitions. Shapes


FIGURE 4. A diagrammatic illustration of the construction forcorneal shapes 5 to 7, representing corneas after refractivesurgery. The central region is a circular segment with aninstantaneous power of 40 D. The outer segment is alsocircular with its center, Q>, on the central axis (h = 0) andwith an instantaneous power of 44 D (the curves are notdrawn to scale). There is an intermediate zone whose poweris 193.5 D, but its width of 0.06 mm is too small to be shown.The central vertical line is the reference axis for shape 5.Shapes 6 and 7 are the same as shape 5, except that thecornea is decentered so that the reference axis then occursat S(1 or S7. All three reference lines pass through the centerof the inner circle, C,, and are thus normal to the cornealsurface.

10 and 11 are similar to shape 9, but the reference axisis shifted. Shape 12 has a peripheral circular segmentsmoothly joined to a central segment with axial andinstantaneous powers that are polynomial functionsOf)'.

INSTANTANEOUS AND AXIAL POWERSOF THE 12 SHAPES

Shapes 1 to 4The first four corneal shapes are conies with eccentric-ities of e = 0 (circle), 0.5 (ellipse), \/n = 1/1.3375(ellipse with zero refractive spherical aberration), and1.0 (parabola). The ellipse is an adequate first-orderapproximation of the corneal shape (in a meridionalplane) in which the average cornea is represented byan eccentricity of approximately 0.5 and the range isnearly 0 to 1.

The axial and instantaneous powers are":

where Pu is the paraxial power (at )' = 0) and whereris the paraxial radius of curvature. The position z foran ellipse will be given in equation 33. The connectionbetween P() and ris given by equation 19:

Pt) = (n- \)/r. (19)

Our hypothetical corneas assume P{) = 44 D (r =7.(5705 mm), which is a value close to that for normalhuman corneas. The plot of z versus y is shown in thetop panel of Figure 7 for e = 0 (dotted-dashed line)and e = 1 (solid line). The top panel shows how smallthe difference is between a circle (e = 0) and a para-bola (e= 1) as long as one stays within several millime-ters of the reference axis.

For small values of ey/r, the square root in equa-tions 17 and 18 can be expanded in a Taylor expan-sion. It is found that the deviation of instantaneouspower from the paraxial power is three times the devi-

P,(y) = /V

(17)

(18)

FIGURE 5. A diagrammatic illustration of the construction forthe segments of corneal shape 8. The central region is acircular segment with an instantaneous power of 40 D, andthe outer segment has an instantaneous power of 44 D (notdrawn to scale). The outer segment is also circular. At thetransition point, the two centers of curvature, C| and C2, liein the same direction, normal to the cornea, so that thereis a smooth tiansition between ihe two segments.


ation of the axial power. For axially symmetric systems,this 3:1 ratio is similar to the deviation in refractivepower in the tangential and sagittal planes.

Shapes 5 to 7

These three shapes are for corneas chat represent con-figurations alter refractive surgery. They have a centralcircular zone of constant instantaneous power {l\ =40 D) surrounded by an outer circular zone of higherconstant instantaneous power (P> = 44 D). For shape5, both these circular segments are chosen to havetheir centers on the optic axis. Shapes 6 and 7 areidentical to shape 5, but the reference axis is displaced(Fig. 7, middle curve). Figure 4 shows that where thetwo circles join, there is a sharp change in cornealslope. If this change occurred at a single point, thecorneal curvature at that point would be infinite. Toavoid infinite quantities, we use a very small circulartransition zone.

FIGURE 6. A diagrammatic illustration of the construction forcorneal shapes 9 to 11, representing a kcraioconic corneawith different cone positions. The central region is a circularsegment with an instantaneous power of 52 D. The outersegment is also circular, with its center on the central axis(h = 0) and with an instantaneous power of 44 D (for claritythe diagram is not drawn to scale). The transition regionhas an instantaneous power of 36 D. The central1 verticalline is the reference axis for shape 9 in which the cone iscentered. Shapes 10 and 1 1 are similar to shape 9, but thereference axis is decentered (not shown). The referenceline for shape 10 passes through the center of curvature ofthe inner circle. The reference line for shape 11 passesthrough the center of curvature of the outer circle. All refer-ence lines are normal to the corneal surface.

- 2 •

shapes 1 and 4

reference axes for shapes:

61 5| 71

reference axes for shapes:

11/ 10

- 4 - 3 - 2 - 1 0 1 2 3 4corneal distance from axis (mm)

FIGURE 7. Several of the shapes we use are drawn to scale.The lop curve shows the z positions of shape I (n = 0, asphere shown by dotted—dashed line) and shape 4 (n = 1,a parabola shown by solid line), with both having pai axialpower of 44 D. The difference between these two shapes isvery small within the firsjt few millimeters from the axis.Shapes 2 and 3 have positions between shapes 1 and 4. Themiddle curve shows shapes 5 to 7 as the solid line that showsa slight flattening in the central region. The dotted-dashedline is for a sphere of 44 D (the same as the dotted-dashedline of the top curve). Shapes 5, 6, and 7 diller only by thelocation of the reference axes, as shown. The bottom curveshows shapes 9 to 11 as the solid line. The dotted-dashedline is again a sphere of 44 D. The difference between thesolid and dotted-dashed lines are too small to be seen.Shapes 9 to 11 are not identical, but the differences aremuch loo small to be seen. Shapes 9,10, and 11 differ mainlyby the location of the reference axes, as shown.

To gain insight regarding the properties of circu-lar segments, shapes 5 to 7 will be discussed quantita-tively. In shape 5, the reference axis is identical to theoptic axis (the axis containing the centers of curvatureof both inner and outer circles). The angle of thenormal at the two transition points is chosen to be (9,= 0.3 and 0^ = 0.34 radians. The inner and outerzones have radii of r, = (n - \)/P\ = 8.4375 mm andr-2 (n — \)/P> = 7.6705 mm. The distance of the twotransition points from the optic axis is calculated tobe )') = r, sin(0|) = 2.4935 mm and y, = r-, sin(6^) =2.5580 mm. The difference between the distances y,and )'i is:

= Jfc - >'i = 0.0645 mm. (20)

The instantaneous power of the transition zone is

P, = AiyP

= 198.5 diopters. (2/)


Equation 21 is obtained by integrating equation6 across the transition /one. Equation 21 also was pre-sented as equation 22 of Klein and Mandcll.10 Equa-tion 21 can be rewritten in a more instructive manneras:

has higher power. The center of curvature of the outerzone lies at a height of

P, = /J,w,. (22)

where PlltWh = (V>, + P,)/2 = 42 D is the average ofthe initial and final axial powers, and )'AVi. = (>'i + f>)/2 = 2.52575 mm is the average of the two transitionpoints. The corresponding differences are A)1, whichis given in equation 20, and APn = 4 D. The precisesize of the transition zone, A)1, is not critical to any ofour calculations. As long as /JMVK, >'AVI., and A/', arespecified, the transition A}> can be any small valueas long as the instantaneous power, /-*,, takes on thecorresponding value as specified by equation 24. Inparticular, as Ay goes to zero, P, goes to infinity.

Shapes 6 and 7 have the reference axes displacedby —0.1 and +0.1 radians, respectively, from the opticaxis (Fig. 7). The angle of the normals at the twotransition points for shape 6 are 6\ = 0.3 + 0.1 = 0.4and #_, = 0.34 + 0.1 = 0.44. The instantaneous powersof the three zones are unchanged because instanta-neous power is independent of axis. All that remainsto be calculated for specifying shape 6 are the new yvalues of the transition points. The first transitionpoint is )'| = r, sin(0|) = 3.2857 mm. The secondtransition point must be calculated differently becausethe center of curvature of the outer zone no longerlies on the new reference axis (S,, in Fig. 4). The differ-ence in )' positions can be calculated from the radiusof curvature of the transition region, r,;

y-i - )'i = r,|>in(02) - sin(0,)l = 0.0621 mm (23)

where r, = (n — \)/Pt. Thus, y, is found to be 3.3478mm. A similar calculation with 9, = 0.3 - 0.1 and 0?= 0.34 - 0.1 is used to define shape 7.

Shape 8

The instantaneous powers for shape 8 are the same asthose of shape 5 except that there is no high curvaturetransition zone. The instantaneous powers of 40 Dand 44 D meet at the transition point whose }• positionis:

y, = r, sin(0,) = 2.6542 mm (24)

where 0, = 0.32 radians is the average of the two transi-tion angles of shape 5. The center of curvature of theinner zone must lie on the reference axis, but thecenter of the outer zone is above the axis because it

h = (r, - r2) sin(0,) = 0.2413 mm (25)

above the axis. If the cornea were a surface of revolu-tion, the central zone would be spherical and theouter zone would be toric.

The above discussion has specified the instanta-neous powers of shapes 5 to 8. For the forthcomingplots, (see Results) we calculated the axial powers bynumerical integration using equation 7. However, be-cause all the segments are circular, the axial powerhas a simple analytic expression given by equation 19of Klein and Mandcll10:

(26)= P,(\ ~ h/y)

where h is the distance from the axis to the center ofcurvature of the circular segment. For shape 8, theheight is given by equation 25. For shape 5, the heightof the outer segment is h = 0 because it was chosento be on the reference axis.

A general method for calculating h is to use equa-tion 9, which can be rewritten in the form:

i\(y) = P,(y)U - - J [i - P.(y')/P,(y)w\. (27)0

By comparing equations 26 and 27, we find:

h= [1 - r,(y')/P,(y)]dy'. (28)J(I

The value of h is a constant when y is in the rangeof the last spherical segment because the integrandvanishes there. Because the transition points areknown for shapes 5 to 8 and because P\(y) is constantbetween transition points, it is straightforward to useequations 26 and 28 to calculate axial power. Withshape 6, for example, the contribution to the integralfrom the central zone is (1 - 40/44) 3.2857 = 0.29091mm; the contribution from the transition zone is (1- 198.5/44).0621 = -.21806 mm, and there is nocontribution from the outer zone. Thus, h = 0.29091- 0.21806 = 0.07285 mm for y in the outer zone, andthe axial power based on equation 26 is: Ptl(y) = 44+ 3.21/}-D.

Shapes 9 to 11

These shapes are meant to represent keratoconus withthe cone in three different positions. The shapes con-sist of smoothly joined segments of circles—i.e., seg-


ments with constant instantaneous power, (see Fig.6)—but they differ from shapes 5 to 7 in that thecentral curvature is larger than the outer curvature,and the transition curvature is lower than either ofthe other curvatures. The outer segment is 44 I), thesame as in shapes 5 to 8. For shape 9, the referenceaxis is aligned with the optic axis such that the outercircle has its center on the reference axis.

The instantaneous powers for the three segmentsof the hypothetical conical shapes are:

P, = 52 D for \y-a\ == 0.5 mm

= 36 D for 0.5 mm < \y - a\ < 1.0 mm (29)

and

= 44 D for 1.0 mm < \y - a\

where the displacement, a, is the ^location of thesymmetry point of instantaneous power. For shapes 9,10, and 11, the value of a is taken to be 0, 0.5, and1.5 mm, respectively (Fig. 7). Note that shapes 9, 10,and 11 arc almost the same shape with differentchoices of reference axes. If they had been exactly thesame shape, the separation between transition pointswould not be fixed at 0.5 mm as specified by equation29; rather, the separation would change slightly as thereference axis shifted and the cornea rotated. Thedependence of the separation of transition point onthe reference axis was seen in equations 20 and 23 forshapes 5 to 7. We prefer to allow shapes 9 to 11 tobe slightly different from each other to allow theirspecification by equation 29 to be simple.

The dioptric value and zone size of the 36 1) transi-tion zone in equation 29 was not chosen arbitrarily.We wanted the center of curvature of the outer zone(the 44 D region) to lie on the optic axis when a =0. This condition forces Plt = P, for the outer zone.According to equation 9, the integrated deviation ofpower above 44 D in the central zone must equal theintegrated power in the transition /one. This condi-tion forces the width of the transition zone to be Aiv =(52 - 44) X 0.5/(44 - 36) = 0.5 mm. The conditionspecified by equation 9 guarantees that as long as kcra-toconus involves a bulging on top of a normal cornea,the central region of the conus with higher than aver-age power must always be surrounded by a balancingregion of lower than average power.

Shape 12

Shape 12 (not illustrated) represents a simulation ofkeratoconus similar to shapes 9 to 1 I but with continu-ously varying instantaneous power for the centralzone, represented by a polynomial in even powers of)'. As with shapes 9 to 11, the cone will have a radiusof 1 mm; outside the cone, the instantaneous power

is 44 D. We chose even powers so that the corneawould be the same for positive and negative values ofy. The axial power for the central zone was chosen tobe of the form:

P,,(y) = 52 + //f + cf for 1 mm. (30)

The value of 52 was chosen so that the axial andinstantaneous power at the apex would be 52 D, asfor shapes 9 to 11. The two conditions on the coeffi-cients a and ft arc that the value and first derivative ofthe axial power must smoothly blend into the constantoutside power of 44 D at |)>| = 1 mm. Thus, P,,(\) =44 and the first derivative must vanish (/J,,(l) = 0)for the instantaneous power to be continuous. Theseconditions uniquely determine Pn to be:

P,,(y) = 4 4 + 8(1 - / ) ' - for \y\ < 1 mm

= 44 for 1 mm.(31)

From equation 6, the instantaneous power is:

P,(y) = 44 + 8(1 — / ) ( ! - 5/) for \y\ < 1 mm

= 44 for 1 mm.(32)

Calculations of Position Power, Pp, for the 12 Shapes.To use equation 14 to calculate the position-basedpower, Pp, we must first calculate z(y) that specificsthe conical position in Cartesian coordinates.

For shapes 1 to 4, which are ellipses, the positionis

z = \r - \r - (1 - * " «2). {33)

For ellipsoids, this value for z can be substitutedinto equation 14 to give the formula:

P,,= (n- \)/[r (ez/r)~Vn. (34)

Note the similarity to equation 17.For shapes 5 to 11 in which the instantaneous

power is piecewise constant, the cornea is made up ofa series of circular arcs. To specify the location z(y),we developed an algorithm for locating each of thecircular arcs. Assume that the radius of curvature, rr

and the end position of each arc are known. The cen-ter of curvature of the central arc is located at (/t,, Z\)= (0, r,), where z, is the z position of the center ofthe ilh arc.

The problem can be stated as follows: given rp yn

hj, and z} for the ji.li arc, calculate the location of thecenter of curvature of the next arc (/»,., / ; zl+i). From


FIGURE8. The top panel is a diagram relevant to the iterativecalculation of the center of cui"vature for a cornea com-prised of circular arcs. The conical slope with reference toI he vertical is tan(0;), where Ht is the angle that the conicalnormal makes with the reference axis at the end of the_/tharc. The bottom panel is a diagram showing how to calculateconical position and axial distance for a cornea comprisedof circular arcs when the center of curvature is known. Theinstantaneous center of curvature is a distance, /*,, above thereference axis for the \th arc. Angle 0 is the angle of inci-dence of an incoming ray, parallel to the reference axis,that strikes the cornea at a distance y from the optic axis.The distance, d, is the axial distance.

the top panel of Figure 8, it is seen that at the pointof transition between the two arcs, the direction ofthe corneal normal, 0r can be calculated from:

sin(0;) = (>', - h,)/rr {35)

The change in the center of curvature Ah and Azcan be calculated in terms of the change in radius ofcurvature Ar = r/(, — rt (which arc known quantities):

Ah = Arsin(0;)

Az = Arcos(0;).

{36)

{37)

From equations 36 and 37, the new center of cur-vature (/*,., i, z/(,) can be calculated. The z position of

a general point on the yth arc can be calculated finallyin terms of the y position on the yth arc by:

)•) = z, - [rf - (y - V ) \n {38)

The bottom panel of Figure 8 shows that for anarbitrary point, y, within an arc, a connection similarto equation 35 can be made between the radius ofcurvature, rn and the axial distance, d:

sin(0) = {y- h,)/r, = y/d {39)

\/d= (1 - h,/y)/rr {40)

This is the relationship that was also seen in equa-tion 20.

For shape 12, in which the axial and instantaneouspowers are polynomials in the central zone, there isno closed form solution for z{y). In this case, the posi-tion must be calculated by carrying out a numericalintegration of equation 12.

Once z(}) is known, equation 14 can be used tocalculate position power, Pp. For all the shapes we areconsidering, the magnitude of Pp deviates little fromits paraxial value. Therefore, to make the deviationsapparent in the plots, we have multiplied the deviationaway from the paraxial value, P(), by a factor of 5. Thus,the quantity plotted in Figures 9 to 11 is given by:

{41)

shape 1 shape 2

2 3 4 0 1corneal distance Irom axis (mm)

FIGURE!). The instantaneous (/',, thin line), axial (/>„, dashedline), position {!)„ thick line), and refractive powers (Pf, dot-ted-dashed line) for four elliptical corneal shapes are shown.The eccentricities are 0 (a sphere), 0.5 (an eccentricity simi-lar to the human cornea), \/n = 1/1.3375 (an ellipse thathas zero refractive spherical aberration), and 1.0 (a parab-ola). The deviations in Pv have been multiplied fivefold tomake them visible.


Position power is directly related to the fluores-cein pattern because it can be written as

P,,= (n- \)/(r+6) (42)

where r is the paraxial radius and 6 is the deviationfrom paraxial radius that can be displayed as a fluo-rescein pattern. Thus, a contour map of Pv could berelabeled as a contour of the deviation, 8, by solvingequation 42 for 6.

Calculation of Refractive Power for the 12 Shapes.From the derivations of the incident angle 0 in equa-tion 8 and the derivation of z(y) in the precedingsection, it is now possible to calculate P/, the refractivefocal power, that was defined in equation 15.

RESULTS

Shapes 1 to 4Figure 9 shows graphs of the four powers for cornealshapes 1 through 4 plotted for positive values of y.

shape 5 shape 6

• 5 0shape 7

axis offset b 0 1 radians „ -

shape 8

tone periphery (no offset)

corneal distance from axis (mm)

FIGURE 10. This figure is similar to Figure 9 except for thechoice of corneal shapes. The four shapes have a centralcircular zone with a power of P, = /'„ = Pt, = 40 D and anouter circular /one whose instantaneous powei is 44 D.Shapes 5 lo 7 have an intermediate /one with an instanta-neous power of 193.5 D. The magnitude of the transitionpower was fixed by the requirement that the center of curva-ture of both the inner and outer segments of shape 5 shouldlie on the reference axis (see text). Shapes 6 and 7 are thesame as shape 5 except for shifts of the reference axis by—0.1 and +0.1 radians. These plots show that refractivepower (dotted-dashed curve) is always just slightly above theaxial power. The difTerence between axial and refractivepowers is approximately a quadratic function of )'. If theshape of the cornea is changed by laser ablation, one mightprefer to connect the change in shape to the change inrefractive power by using the true index of the cornea (be-cause it is purely the coinea that is changing shape), inwhich case n = 1.376 should be used in equation 2.

55

50

45

40

12 35

offset

V\\

\

, IH. 0 1o

155gE508

45

40

35

offset

I I

shape 9= 0mm

2 3 4

shape 11

= 1 5 mrr

/

"• o - "

I '

shape 10

offset = 0.5 mm

J

350 1 2 3 4 0 1

corneal distance from axis (mm)

FIGURE it. This figure is similar to Figure 9 except that thefour shapes are chosen to simulate keratoconus. Shapes 9to 1 1 are for a coinea with an apical power (1-mni diameter)of 52 D, flanked on both sides by a 0.5 mm /one of 38 D,which is then surrounded with a 44 D outer region. Thethree panels are for three placements of the reference axis.The last shape has a polynomial-dependent 2-mm diameter/one in the center and a 44 D outer region.

The instantaneous, axial, position, and refractive pow-ers are shown as thin, dashed, thick, and clotted-dashed lines, respectively. For shape J, where e = 0(a circular shape), the three shape-based powers areconstant and equal to each other. Only the refractivepower is different. For shape 2, with e = 0.5 (a shapesimilar to the average cornea), the instantaneous andaxial powers decrease while the refractive power in-creases. For shape 4, with e = 1.0 (a parabola), therefractive power joins the others as a declining func-tion Of)!.

Shape 3 is at an intermediate eccentricity givenby e = \/n (a surface known as a Cartesian Oval) forwhich there is zero refractive spherical aberration sothat Pj is a constant independent of)1. This result canbe derived by equating the optical path lengths fortwo rays to reach the focal point:

where nf is the optical path length (the geometriclength times the index of refraction) for a ray at y =0 to go from the corneal vertex to the focal point.The right side of equation 43 is the optical path lengthfor a parallel ray arriving a distance y from the optic;axis. Equation 43 can be rewritten as:

z = - l/rr)J/2r (44)

which is the equation of an ellipse with eccentricity e.= \/n.


The position power is relatively constant. As anexample, for shape 4, the position power seems tohave a value of approximately 42 D at y = 4 mm.However, the fivefold amplification shown in equation41 means that the true position power at y = 4 is: 44+ (42 - 44)/5 = 43.6 D, which is barely changedfrom the paraxial value of 44 D.

Shapes 5 to 8

Shapes 5 to 7 represent the case of excimer laser cor-neal flattening. The four powers are shown in Figure10. The inner and outer zones have instantaneouspowers of 40 D and 44 D. The transition zone has P,= 198.5 D. In shape 5, both inner and outer circleshave their centers on the reference axis, correspond-ing to spherical corneas. In the inner and outer zones,the axial power equals the instantaneous power.Shapes 6 and 7 have the same cornea! shape that shape5 has, but the reference axis lias been shifted down-ward and upward by 0.1 radians. The transition zonechanges size because of the foreshortening caused bythe rotation as discussed in Methods. Because shapes6 and 7 are produced by equal and opposite rotations,they can be interpreted as views of the identical corneaalong meridia that are 180° apart. One of the im-portant features that can seen in shape 7 is that as thereference axis gets closer to the tiansition zone, thejump in axial power at the transition becomes largerthan the jump in instantaneous power. In this case,the axial power jumps to approximately 46 D. If thereference axis were shifted to the beginning of thetransition zone, the axial power would jump to 198.5D. This illustrates how it is possible for axial power toseem to have a more dramatic change than instanta-neous power. In corneas that have undergone photorefractive keratectomy (excirner flattening), one canget unusually large readings of axial power by themechanism shown in shape 7.

Shape 8 has the same inner zone and outer zoneinstantaneous powers as shape 5. It does not have thevery high curvature Lransiiion zone used in shape 5 toallow the outer zone to have its center of curvatureon the reference axis. Shape 8 has a discontinuity ininstantaneous power, whereas the other three powersare continuous. In each of the shapes 5 to 8, the dis-continuity of refractive power is similar to that foundfor axial power, one of the important findings of ourstudy. The change in position power is small, just asfor shapes 1 to 4, when one takes the fivefold magnifi-cation into account.

Shapes 9 to 12Shapes 9 to 12 represent two approaches for model-ing kcratoconus. P, for shape 12 is a smooth polyno-mial joined to a spherical base. Shapes 9 to 11, onthe other hand, consist of discrete circular segments

with abrupt discontinuities in P, at the transitionpoints. The central segment representing the apexoi the cone in shapes 9 to 11 has a diameter of 1 mmand a power of P, = 52 D. The central segment isflanked by a 0.5-mm segment with P, = 36 D, whichin turn is surrounded by a spherical zone of 44 D.Several interesting features can be observed in theseplots. In shape 9, the reference axis is centered atthe apex of the cone. Although /-", has the thin ringof 36 I), the axial power, Pfl, never is less than 44 D.The same is true for shape 10, in which case the axisis shifted downward by 0.5 mm. In shape 11, thereference axis is shifted downward by 1.5 mm so thatthe full cone is shifted to positive values of}'. In thiscase, /*„ bears little resemblance to P,. In fact, in theregion between y = 1 and 2 mm, where the peak ofthe cone is located (/-", = 52 D), the axial power shiftsfrom 40 D to 46 D. Thus, it is seen that axial powercan severely distort the true corneal shape. Shape 12is a smooth version of shape 9.

An important feature of the 12 shapes is the closeconnection between the refractive power and the axialpower. This is especially apparent for those shapesthat have abrupt discontinuities in instantaneouspower. This connection is understandable because P,,is really a measure of corneal slope, just as is P/.Whereas the instantaneous power can show dramatic,large deviations from axial power, the refractive poweris always just slightly above the axial power.

We find that a good approximation to refractivepower is:

P, = />,(! +Ky*) {45)

where K ranges from approximately 0.005 to 0.009(for y measured in mm) for all the shapes we haveconsidered. The largest value we found occurs for asphere. The value of K is related to the spherical aber-ration of the corneal shape.

One final point that can be seen based on viewingthe 12 shapes is that whenever /-*„ is a constant, it isalso equal to P,. This result also can be seen directlyfrom equation 6.

DISCUSSION

Roberts1' compared Pn, P,, and P, for ellipsoids with e= 0, 0.3, and 0.5. For these cases, Pt increased as afunction of}1, whereas /-*„ and P, were either constant(for e = 0) or were decreasing functions of y. Wehave (bund that the relationship for Pf is reversed forellipses of higher eccentricity, representing a signifi-cant proportion of corneas. In addition, we found thatfor various corneal shapes with dramatic changes ininstantaneous power, the refractive power, Pt, more


closely follows P,, than P, or Ptl. The spherical aberra-tion deviation between /'and Pn can be approximatelyaccounted for by equation 45. At a radius of 2 nun (arelatively large pupil size), the most extreme differ-ence between Pf and P,, (occurring for shape 1, thesphere) is approximately 1.6 D. The deviation of thecorrect value of P, from the value given by equation45 (with K = 0.007) is less than 0.4 D. Thus, presentvidcokcratographs that report P,, do provide a nearapproximation of P, (especially if the correction termgiven by equation 45 is used). This discussion is notmeant to be interpreted as our advocating the use ofaxial power to represent refractive power. In equations10 to 16, we have shown how exact refractive powercan be derived from axial power so there is no needto use unmodified axial power for that purpose. Theplots in Figures 10 and 11 demonstrate that instanta-neous power grossly can misrepresent the refractivepower. Axial power, on the other hand, when usedwith K = 0.007, gives a very close approximation torefractive power.

Present videokeratographs perform their calcula-tions one meridian at a time. This procedure is accept-able for axially symmetric corneas. However, for cor-neas that are not axially symmetric, such as those withastigmatism or keratoconus, present algorithms forcalculating corneal shape have systematic errors be-cause of their incorrect assumption that rays remainin the meridional plane. Consider the chief ray thatgoes through the camera lens. It enters the camera inthe meridional plane (by definition of its being a chiefray). However, before reflecting from the cornea, itwas not in the meridional plane if the normal to thecornea was not in the meridional plane (as wouldoccur with astigmatism). To obtain the correct cornealshape in this situation, a new algorithm is neededthat calculates bias-free corneal shape (no systematicerrors) even for corneas without axial symmetry.18 Forthese corneas, multiple definitions of Ptl and P/ arepossible because the normal to the surface does notintersect the optic axis. If the axial distance and focallength distance are redefined in terms of the point ofclosest approach to the optic axis, we conjecture thatall results of the current article will still be approxi-mately correct (correct to within a few percentagepoints for corneal shapes found in the population).

Relative Merits of the Four Definitionsof Power

Corneal shape can be represented by the curvature(related to P,), the slope (related to Plt), or directly bythe z value (related to Pfl). In this discussion, we con-sider the advantages and disadvantages of the threeshape-based powers and also refractive power for gen-eral (not necessarily axially symmetric) corneal

shapes. It is likely that each will continue to be usedfor specific applications.

Instantaneous Power (Representing the Second Deriva-tive of Corneal Shape). Advantages. An advantage of in-stantaneous power is that it is directly proportional tothe local curvature and thus represents an appropriaterepresentation of local shape. It is useful in the diag-nosis of keratoconus and other corneal anomalies thathave local deviations away from the normal range.Instantaneous power is based on the second derivativeof the corneal shape function and is highly sensitiveto local changes in shape. This can be an importantadvantage when a clinician wants to visualize localanomalies. Shapes 9 to 12 of Figure 11 show how P,offers a much more dramatic representation of a cor-neal "bump" than does Pn.

The instantaneous power that we have been dis-cussing in this article is not intrinsic to the surfacebecause it depends on the location of the axis. A usefulshape descriptor is the Gaussian curvature, which isthe product of the two instantaneous curvatures. Inkeeping with our desire to report quantities in dioptricunits, one could define "Gaussian power" as the geo-metric mean of the maximum and minimum powersat each point. This definition is especially useful forsoft contact lenses because the Gaussian power wouldbe constant as the lens flexes (a property of Gaussiancurvature).

Disadvantages. A disadvantage of instantaneouspower is that it usually is not measured directly andmust be calculated from axial power by taking a deriva-tive (see equation 6), which can be a problem. Indealing with noisy data, derivatives will exaggerate thenoise; smoothing the axial power may be necessarybefore or after the derivative is taken. Smoothing canbe achieved by using a spline, but too much smoothingcan diminish features that arc present.18 The y depen-dence in equation 6 indicates that the effect of noisewill be greatest at the periphery of the cornea, wherey is largest. Near the center of the cornea, where y issmall, P, will be very close to /-*„ and will be insensitiveto the derivative term.

Although instantaneous power describes local cur-vature, it does not allow direct visualization of theoverall corneal location. The location, z, of each pe-ripheral point depends on the powers between thatpoint and the corneal axis.

Axial Power (Representing the First Derivative of Cor-neal Shape). Advantages. Axial power is the power thatis measured directly from vidcokcratography. The rawdata from a videokeratograph are related closely tothe corneal slope because it is slope that controls thedirection of reflected rays. Slope is directly relatedto axial power by equation 10. Because of this directconnection to the raw data, axial power is most robustto noise, which is a practical justification for its use.

2108 Investigative Ophthalmology 8c Visual Science, September 1995, Vol. 36, No. 10

From axial power, it is easy to convert to the othertypes of power. Instantaneous power is found by takinga derivative (equation 6), position power is obtainedfrom equation 12, and refractive power is found byusing Snell's Law or by the approximate formula givenin equation 45.

Disadvantages. The main disadvantage of axialpower is that it is not intrinsic to the local surface.Axial power is dependent on the choice made forthe reference axis, which is arbitrary. The connectionbetween position (z) and axial power is not directbecause z depends on all values of /'„ from the vertexto the point of interest, as it does for Pt, but it is notas extreme because only one rather than two integra-tions arc needed.

Position Power (Representing the Oth Derivative of Cor-neal Surface Shape). Advantages. Although instanta-neous and axial powers are used clinically as descrip-tors of corneal shape, neither allows direct visualiza-tion of corneal topography, as would be achieved bya specification of their x, y, and z coordinates. Whenconsidering general, nonaxially symmetric surfaces,this representation has the decided advantage of con-veying the full information about the cornea in a mosteconomical manner. A slope-based representation re-quires two numbers to represent slope at each point(dz/dx and dz/dy). A curvature-based representationrequires three numbers at each point (sphere, cylin-der, and angle). The position representation, on theother hand, requires only a single number, dp, at eachpoint. A final advantage is that for severely degradedcorneas or for the sclera, it is not possible to obtainhigh-quality reflected images so the Purkinje imagemethods used to get slope directly are not possible.In those cases, it is desirable to measure position,rather than slope, directly.

In Figures 9 to 11, we have displayed the recipro-cal of the corneal distance from the paraxial centerof curvature in terms of the position power, P(l. Thecorneal distance, df,, also can be expressed as the par-axial radius of curvature plus the distance of the cor-nea from the reference sphere, centered at the para-xial center of curvature. The concept of a referencesphere is in effect achieved by the clinical test of thefluorescein pattern. The fluorcscein pattern repre-sents the variation in the separation between cornealpoints and the posterior surface of a rigid contactlens. Unfortunately, the actual fluorcscein test is onlyqualitative, even though it provides a very sensitivemeasure. However, simulations of the fluorescein pat-tern in computer displays are quantitative and can beused to visualize corneal shape directly and effectively.

Disadvantages. The disadvantage of the coordinatesystem representation is the accuracy needed in mea-surements of corneal position to calculate Pn, P,, orPf. To obtain these quantities, derivatives of corneal

position must be taken, which requires siibiiiicron ac-curacy in the value of z or a representation of z thatenforces smoothness.11 A second problem with thecoordinate system representation is that interestingcorneal features such as keratoconus do not stand out.

Refractive Power (Representing the Refractive Prop-erty of the Cornea). Advantages. The principal advantageof refractive power is that for any corneal point, itindicates the local power change in the cornea neededto correct for a given refractive error. This allows clini-cal application whereby it can be determined if thecorneal change produced by refractive surgerymatched the one desired. Refractive power is also theonly power that has application to the calculation ofthe contribution of corneal power to image quality. Ifit is desired to correct for the aberrations of the eye,the appropriate corneal changes must be made interms of refractive power.

Disadvantages. Throughout this article the eye wasassumed to consist of a single refractirrg surface. Theposterior surface of the cornea arrd the crystalline lenswere ignored. Although the cornea is the dominantcontributor to refractive power, a proper calculationof refractive power requires raytracing that assumeskrrowledge of the locations and indices of refractionof the multiple ocular surfaces. At present, it is diffi-cult to obtain the required information. However, inthe future the refractive power across the corrrea willbe assessed directly by measuring the wave front aber-rations. When the wave front aberrations are coupledwith information about corneal shape, it will be possi-ble to predict accurately how much of the cornea mustbe ablated to minimize the aberrations of the eye.

CONCLUSION

Each of the four definitions of corneal power has advan-tages and disadvantages for specific applications. It issometimes of value to have available more than one ofthe four power expressions to describe different featuresof corneal topography. It is anticipated that each of thefour powers as defined will continue to be useful as anexpression of the corneal topography.

Key Words

cornea, corneal power, corneal topography, videokeralogra-phy

Acknowledgments

The authors thank Brian Barsky for discussions of theseissues.

References

1. Roberts C. The accuracy of 'power' maps to displaycurvature data in corneal topography systems. InvestOfjhthahnol Vis Sa. 1994; 35:3525-3532.


2. Mandell RB. Conical power correction factor for pho-torelractive keraiectoniy. / lie/rod Corveal Sing. 1994;10:125-128.

3. Mandell RB, Klein SA, Shie CH, Barsky BA, Yang Z.Axial and instantaneous radii in videokeralography.ARVO Abstracts. Invest Ophthalmol Vis Sci. 1994; 35:2079.

4. von Hclinholu H. liandbuch dei I'hysiologischen OptikVol. 2. Haniburg/Leip/ig: Verlag von Leopold Voss;1896:16-18.

5. Mandell RB, St. Helen R. Mathematical model of theconical contour. BrJ rhysiol Opt. 1971;26:183- 197.

6. Klyce SD. Computer-assisted conical topography:High-resolution graphic presentation and analysis ofkeratoscopy. Invest Ophthalrnol Vis Sci. 1984; 25:1426—1435.

7. Klyce SD, Dingeldcin SA. Conical topography. In:Masters B. Non-invasive diagnostic techniques in Ophthal-mology. New York: Springer-Verlag; 1990:61-77.

8. McDonnell PJ, Garbus |, Lopez PF. Topogiaphic anal-ysis and visual acuity after radial keratoiomy. AmJOphthalmol. 1989; 106:692-695.

9. Maguire L), Bourne WM. Conical topography of earlykeratoconus. Am J Ophthalmol. 1989; 108:107-112.

10. Klein SA, Mandell RB. Axial and instantaneous power

conversion in conical topography. Invest OphthalmolVis SCI. 1995; 36:2155-2159. '

11. Doss ED, HuLson RL, Rowsey IJ, Brown R. Method forcalculation of conical profile and power distribution.Arch Ophthalmol. 1981 ;99:1261-1265.

12. Wang J, Rice D, Klyce SD. A new reconstruction algo-rithm for improvement of conical topographical anal-ysis. / Refract Corneal Surg. 1989;5:379-387.

13. van Saarloos PP, Constable IJ. Improved method forcalculation of corneal topography for any photokera-toscope geometry. Opt Vis Sa. 1991;68:960-966.

14. Klein SA. A corneal topography algorithm that pro-duces continuous curvature. Optom Vis Sci. 1992;69:829-834.

15. Mandell RB. The enigma of the corneal contour.CIAOJ. 1992; 18:267-273.

16. Bibby MM, Townsley MG. Analysis and description ofconical shape. CL Forum. 1976; 1:27-35.

17. Roberts C. Characterization of the inherent error ina spherically-biased conical topography system inmapping a radially asphcric surface. / Refract Cor Surg.1994; 10:103-111.

18. Klein SA, Halstcad M, Mandell RB, Barsky BA. Cor-neal topography for general surfaces. ARVO Abstracts.Invest Ophthalmol Vis Sa. 1994; 35:2079.

19. Mandell RB. Profile methods of measuring conicalcurvature./ Am Optom Assn. 1961; 32:627-631.

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