Calculation of the total corneal astigmatism using the virtual cross cylinder method on the secondary principal plane of the cornea

Abstract

This study aimed to establish a virtual cross cylinder method to calculate the total corneal astigmatism by combining the anterior and posterior corneal astigmatism on the secondary principal plane of the cornea based on Gaussian optics. The meridian with the least refractive power, namely, the flattest meridian of the virtual cross cylinder of a ± 0.5 × C diopter, is set as the reference meridian, and the power (F) at an angle of φ between an arbitrary meridian and the reference meridian is defined as F(φ) = - 0.5 × C × cos2φ. The magnitude and axis of the total corneal astigmatism were calculated by applying trigonometric functions and the atan2 function based on the combination of the virtual cross cylinders of the anterior corneal astigmatism and the posterior corneal astigmatism. To verify the performance of the virtual cross cylinder method, a verification experiment with two Jackson cross cylinders and a lensmeter was performed, and the measured and calculated values were compared. The limit of the natural domain of the arctangent function is circumvented by using the atan2 function. The magnitude and axis of the total corneal astigmatism are determined through generalized mathematical expressions. The verification experiment results showed good agreement between the measured and calculated values. Compared to the vector analysis method, the virtual cross cylinder method is mathematically sound and straightforward. A novel technique for calculating total corneal astigmatism, the virtual cross cylinder method, was developed and verified.

Figure 1

A virtual cross cylinder with a ± 0.5 × C diopter, depicting the power profile with a color coded map (a) and a graphical display (b) of the power F(φ). C: power corresponding to the astigmatic magnitude.

Figure 2

(a) A virtual cross cylinder for anterior corneal astigmatism with axis α. The power profile is expressed as F_ACA(φ) =  − 0.5 × ACA × cos2(φ – α). (b) A virtual cross cylinder for posterior corneal astigmatism with axis β. The power profile is expressed as F_PCA(φ) =  − 0.5 × PCA × cos2(φ – β). ACA magnitude of the anterior corneal astigmatism, PCA magnitude of the posterior corneal astigmatism.

Figure 3

(a) Graphical display of the relationship between θ and $\tan \theta$= a/b. a = ACA × sin2α + PCA × sin2β, b = ACA × cos2α + PCA × cos2β; (b) Graphical display of the relationship between a/b and θ = arctan(a/b). The natural domain of the arctangent function is – π/2 < θ < π/2. (c) Application of the atan2 function. If b > 0, θ is in the first or fourth quadrants and thus within the natural domain of the arctangent function, so θ_ATAN2 = arctan(a/b) + 0. If b < 0 and a ≥ 0, θ is in the second quadrant, which is beyond the natural domain of the arctangent function, so θ_ATAN2 = arctan(a/b) + π. If b < 0 and a < 0, θ is in the third quadrant, which is beyond the natural domain of the arctangent function, so θ_ATAN2 = arctan(a/b) − π. If b = 0 and a > 0, θ_ATAN2 = π/2. If b = 0 and a < 0, θ_ATAN2 =  − π/2. (d) Axis angle σ of the total corneal astigmatism, which is determined based on the combination of the signs of a and b.

Figure 4

The two Jackson cross cylinders used for the verification experiment: ± 1.00 D (S – 1.00 D = C + 2.00 D Ax 0) and ± 0.50 D (S – 0.50 D = C + 1.00 D Ax 0). D diopter, Ax axis.

Figure 5

Plots of the lensmeter-measured and virtual cross cylinder method-calculated values with two Jackson cross cylinders of ± 1.00 D and ± 0.50 D. (a) Astigmatic axis. (b) Astigmatic magnitude. The oblique line in the graph is y = x. D diopter.

Figure 6

A spherocylindrical lens of Cy + C Ax 0, depicting the power profile with a color coded map (a) and a graphical display (b) of the power F(φ). C power of the spherocylindrical lens.