The Transposition Calculator

📖 How to Transpose a Prescription

Sphero-cylindrical transposition converts a prescription between its two mathematically equivalent forms — one with a plus cylinder and one with a minus cylinder. Both describe the same optical lens; they are simply different notations.

The Three Steps

1. New Sphere — Add the cylinder power to the sphere power.
2. New Cylinder — Reverse the sign of the cylinder (+ becomes −, − becomes +).
3. New Axis — Change the axis by 90°, staying within 1–180°.

Axis Rules

Optical axes run from 1° to 180° only — zero is recorded as 180°. After adding or subtracting 90°:

• If result is ≤ 0 → add 180° (e.g. −88 + 180 = 92°)
• If result is > 180 → subtract 180° (e.g. 260 − 180 = 80°)
• If result is exactly 0 → write as 180°

Examples:  Axis 10° → 100°  |  Axis 170° → 80°  |  Axis 90° → 180°  |  Axis 2° → 92°

Worked Example

So −2.50 / −1.00 × 90  ↔  −3.50 / +1.00 × 180

Challenge Mode Tips

• Work one step at a time — sphere first, then cylinder, then axis.
• The axis change is the most common source of error; take your time.
• Signs always matter — include + or − on non-zero sphere and cylinder.
• "Plano" (or 0.00) is accepted for any zero power value.
• Difficulty increases level by level — early rounds keep the sphere positive.