A conic section can be described mathematically by Baker’s equation
2 , which solves for
Z: \[Z(x,\ y){=}\ \frac{{-}R{+}\sqrt{{[}R^{2}{-}(1{+}Q)(x^{2}{+}y^{2}){]}}}{(1{+}Q)}\]
where
x and
y are the coordinates on a Cartesian system with the axis of revolution,
R is the apical radius of curvature, and
Q is the asphericity. When
Q < 0, the ellipse is prolate and flattens from the center to the periphery. When
Q = 0, the ellipse is a circle. When
Q > 0, the ellipse is oblate and steepens from the center to the periphery.
When a correction of
D diopters is simulated using
1 of Munnerlyn et al.
6 on a cornea modeled as a conic section of apical radius
R 1 and shape factor
p 1, within an optical zone diameter
S, the resultant curve
Z 2 is derived from
\[Z_{2}(x,\ y){=}Z_{1}(x,\ y){+}dZ(x,\ y)\]
\[Z_{2}(x,\ y){=}\ \frac{{-}R_{1}{+}\sqrt{\ {[}R_{1}^{2}{-}(1{+}Q_{1})(x^{2}{+}y^{2}){]}}}{1{+}Q_{1}}{-}R_{2}{+}R_{1}{+}\sqrt{R_{2}^{2}{-}(x^{2}{+}y^{2})}{-}\sqrt{R_{1}^{2}{-}(x^{2}{+}y^{2})}\]
As shown in a previous analysis of myopia corrections,
5 these equations do not describe a conic section. However, the radius of curvature for each point of the curve
Z 2 is given by
r 2(
x,
y), which can be computed as
\[r_{2}(x,\ y){=}\ {\vert}\ \frac{{[}1{+}(Z{^\prime}_{2})^{2}{]}^{3/2}}{Z{^{\prime\prime}}_{2}}{\vert}\]
After respective first and second derivatives of function
Z 2(
x,
y) are computed and inserted into the formula
r 2(
x,
y), the radius of curvature can be calculated by substituting
Z′
2 and
Z′
2 in
12 .
5 The apical radius of curvature of
Z 2(
x,
y) is
r 2(0). It is calculated by substituting 0 for
Z and leads to the result
\[x{=}0\ \mathrm{and}\ y{=}0,\ r_{2}(0){=}R_{2}\]
Thus, the radius of curvature of
Z 2 at the apex (apical radius of curvature) is the same as the final radius of curvature, which is derived from
1 (Munnerlyn et al.
6 ).
Thus,
Z 2(
x,
y) has an apical radius of curvature
R 2, but its asphericity cannot be computed by the foregoing calculations, because
Z 2(
x,
y) does not describe a conic section. However, a best-fit conic section,
C 2(
x,
y), with apical radius of curvature
R 2 and asphericity
Q 2 can be calculated. We plotted multiple conic sections
C(
x,
y) with asphericity
Q c. Substituting
C 2 for
Z in
9 \[C_{2}(x,\ y){=}\ \frac{R_{2}{-}\sqrt{{[}R_{2}^{2}{-}(1{+}Q_{c})(x^{2}{+}y^{2}){]}}}{(1{+}Q_{c})}\]
To determine the best-fit conic section with shape factor
Q 2, we minimized the sum of the squared residuals
T s(
Q c).
We developed a numerical procedure and performed computations on a computer spreadsheet (Excel 97; Microsoft, Seattle, WA). We defined 31 values for (
x 2 +
y 2)
1/2 ranging from 0 to
S/2 = 3 mm, equally spaced by 0.1 mm. For a given
R 1,
D, and
Q 1,
T s(
Q c) was iteratively calculated for
Q c, ranging from (
Q 1 − 2) to (
Q 1 + 2) in 0.01 increments. Solutions were represented by the
Q c that induced the smallest
T s(
Q c)
2 .
Q c was tabulated for various
D (+1 to +6 D, by 1-D steps),
Q 1 (−0.6–0.4, by 0.2 steps), and
R 1 (6.8–7.8 mm, by 1-mm steps).
We also derived an approximation of
Q 2 in the special case of Munnerlyn-based ablation applied to an aspheric surface using the Taylor series expansion described previously
7 \[dZ{\cong}\ \frac{S^{2}D}{3}{+}\left(\ \frac{3S^{2}}{16R_{1}^{2}}\right)\ \left(\ \frac{S^{2}D}{3}\right)\ {+}\left(\ \frac{3Q_{1}S^{2}}{16R_{1}^{2}}\right)\ \left(\ \frac{S^{2}D}{3}\right)\ {+}\ \frac{dQ.S^{4}}{128R_{1}^{3}}\]
A spherical ablation is equivalent to an aspheric ablation in the special case where Q 1 = 0 and dQ = 0.
The approximation of
8 was equalized to that of a spherical ablation
dZ:
Q 1 =
Q 2 = 0:
\[dZ{\cong}\ \frac{S^{2}D}{3}{+}\left(\ \frac{3S^{2}}{16R_{1}^{2}}\right)\ \left(\ \frac{S^{2}D}{3}\right)\]
Equalizing
15 from
8 gives
\[\left(\ \frac{3Q_{1}S^{2}}{16R_{1}^{2}}\right)\ \left(\ \frac{S^{2}D}{3}\right)\ {+}\ \frac{dQS^{4}}{128R_{1}^{3}}{\cong}0\]
Solving
16 for
dQ results in
\[dQ{\cong}{-}8DR_{1}Q_{1}\]