Power in an Oblique Meridian: The sin² Formula

To find the power of a sphero-cylinder lens in a meridian between its two principal meridians, use F = S + C × sin²θ. S is the sphere, C is the cylinder, and θ is the angle between the cylinder axis and the meridian you want. At 30° from the axis you add a quarter of the cylinder, at 45° half, and at 60° three quarters.

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The ABO Basic exam lists ophthalmic formulas under Ophthalmic Optics (Basic Exam Handbook). This one comes up when a question asks for the power of a cylinder lens in a direction other than its axis.

The formula

F = S + C × sin²θ

• S = sphere power
• C = cylinder power, with its sign
• θ = angle between the cylinder axis and the meridian you want

Along the axis (θ = 0°) you get just the sphere. At 90° from the axis you get the full sphere plus cylinder. In between, sin²θ tells you what share of the cylinder to add.

This is the standard thin-lens model. Strictly, a cylinder lens has no single power in an oblique meridian; the sin²θ value describes the surface curvature there and works as an approximation.

Quick sin² table

Worked examples

Example 1: minus cylinder at 45°

Lens: −2.00 −1.50 × 090. Find the power in the 045 meridian.

1. θ = 90 − 45 = 45°, so sin²θ = 0.50.
2. F = −2.00 + (−1.50 × 0.50) = −2.00 − 0.75 = −2.75 D.

Same lens in plus cylinder: −3.50 +1.50 × 180. θ = 45°, so F = −3.50 + 0.75 = −2.75 D. Same answer.

Example 2: plus cylinder at 60°

Lens: +1.00 +2.00 × 180. Find the power in the 060 meridian.

1. θ = 60 − 0 = 60°, so sin²θ = 0.75.
2. F = +1.00 + (2.00 × 0.75) = +1.00 + 1.50 = +2.50 D.

Example 3: an axis that isn’t on the table

Lens: −1.00 −2.00 × 020. Find the power in the 180 meridian.

1. θ = 20°. sin 20° ≈ 0.342, so sin²θ ≈ 0.117.
2. F = −1.00 + (−2.00 × 0.117) ≈ −1.00 − 0.23 = −1.23 D.

A quick check

The powers in any two meridians 90° apart add up to 2S + C. In Example 3, the 090 meridian is 70° from the axis: sin²70° ≈ 0.883, so F ≈ −1.00 − 1.77 = −2.77 D. Then −1.23 + (−2.77) = −4.00, and 2S + C = −2.00 − 2.00 = −4.00. It checks.

Not sure about cylinder form? Review lens transposition. For prism from decentration, see Prentice’s rule; use the 180-meridian power for horizontal prism and the 090-meridian power for vertical prism (an estimate when the axis is oblique).

Practice problems

1. +3.00 −1.00 × 180, power in the 030 meridian

   Show answer

   θ = 30°, sin² = 0.25. +3.00 + (−1.00 × 0.25) = +2.75 D

2. −4.00 −2.00 × 045, power in the 090 meridian

   Show answer

   θ = 45°, sin² = 0.50. −4.00 + (−2.00 × 0.50) = −5.00 D

3. +2.00 +1.00 × 090, power in the 030 meridian

   Show answer

   θ = 60°, sin² = 0.75. +2.00 + (1.00 × 0.75) = +2.75 D

4. −0.50 −3.00 × 010, power in the 180 meridian

   Show answer

   θ = 10°, sin² ≈ 0.030. −0.50 + (−3.00 × 0.030) = −0.59 D

5. −1.50 −1.00 × 120, power in the 180 meridian

   Show answer

   θ = 60°, sin² = 0.75. −1.50 + (−1.00 × 0.75) = −2.25 D