7.4 Transposition, Spherical Equivalent, Vertex Distance & Prescription Components

Transposition

A sphero-cylindrical lens can be written two equivalent ways — plus cylinder or minus cylinder. Ophthalmology conventionally writes minus cylinder; optometry and many laboratories historically use plus cylinder. Converting between them is transposition, and it is three mechanical steps:

1. Algebraically add the cylinder power to the sphere power — this becomes the new sphere.
2. Change the sign of the cylinder.
3. Rotate the axis by 90 degrees (add 90 if the axis is 90 or less; subtract 90 if the axis is above 90). The result must stay in the 1 to 180 range.

Example 1. +2.00 −1.50 × 180 → sphere +2.00 + (−1.50) = +0.50; cylinder becomes +1.50; axis 180 − 90 = 090. Result: +0.50 +1.50 × 090.

Example 2. −3.25 +1.00 × 045 → sphere −3.25 + 1.00 = −2.25; cylinder −1.00; axis 045 + 90 = 135. Result: −2.25 −1.00 × 135.

Example 3. −1.00 −2.50 × 100 → sphere −3.50; cylinder +2.50; axis 100 − 90 = 010. Result: −3.50 +2.50 × 010.

The two forms describe the same lens: the power in each principal meridian is unchanged. In example 1 the 090 meridian has +2.00 and the 180 meridian has +0.50 in both notations.

Spherical equivalent

The spherical equivalent is the single spherical power whose focus sits at the circle of least confusion — the dioptric midpoint between the two focal lines.

Spherical equivalent = sphere + (cylinder ÷ 2)

Examples.

Uses: ordering a trial soft contact lens for a patient with low astigmatism, estimating a disposable lens power, summarising refractive error in research or screening, and deciding whether astigmatism is low enough to ignore. Limitation: it discards the astigmatic correction entirely, so a patient with significant cylinder will not see as well through the spherical equivalent as through the full correction.

Vertex distance

Vertex distance is the distance from the back surface of the lens to the front of the cornea, conventionally about 12 to 14 mm for spectacles. It matters because the effective power of a lens at the eye changes as the lens moves.

The rule that answers most exam items without arithmetic:

• Minus lenses: moving the lens closer to the eye makes it less effective, so you need more minus. Moving it further away requires less minus.
• Plus lenses: moving the lens closer to the eye makes it more effective, so you need less plus. Moving it further away requires more plus.

A compact mnemonic: as a lens moves away from the eye, effective plus increases.

Formula:

Effective power = F ÷ (1 − d × F)

where F is the lens power in dioptres and d is the distance moved in metres (positive when moving the lens toward the eye).

Vertex conversion in practice

When does it matter? Below about ±4.00 D the change is smaller than the 0.25 D prescribing step and can be ignored. From ±4.00 D upward it must be calculated, and above ±10.00 D it is substantial.

Worked example — spectacle to contact lens, high myope. A −10.00 D spectacle lens at a 12 mm vertex is being converted to a contact lens sitting on the cornea (d = 0.012 m toward the eye):

Effective power = −10.00 ÷ (1 − (0.012 × −10.00)) = −10.00 ÷ (1 + 0.12) = −10.00 ÷ 1.12 = −8.93 D, prescribed as −8.75 D or −9.00 D.

So the contact lens needs less minus than the spectacle, consistent with the rule.

Worked example — high hyperope. A +12.00 D spectacle at 12 mm converted to a contact lens:

Effective power = +12.00 ÷ (1 − (0.012 × 12.00)) = 12.00 ÷ (1 − 0.144) = 12.00 ÷ 0.856 = +14.02 D, prescribed as +14.00 D.

The contact lens needs more plus. This is why aphakic contact lens powers are so much higher than aphakic spectacle powers.

Approximate magnitudes to recognise on sight:

Measuring vertex distance. Use a distometer (vertexometer): the patient closes the eye, the instrument's caliper foot rests on the closed lid and the other arm on the back of the lens, and the scale reads the distance with an allowance for lid thickness built in. Alternatively, view the patient in profile through the slit lamp or use a millimetre rule from the side. Record the vertex distance whenever a refraction exceeds ±4.00 D, because the laboratory must know the distance the power was measured at.

Sphero-cylindrical prescriptions must be converted meridian by meridian when accuracy matters, because the two principal meridians have different powers and therefore different vertex corrections. For a −8.00 −2.00 × 180, convert −8.00 and −10.00 separately and then rebuild the sphero-cylinder.

Components of a spectacle prescription

The blueprint names "prescriptions" as an Optics and Spectacles content item. A complete written spectacle prescription contains:

1. Patient name and date of the examination
2. Right eye (OD) and left eye (OS) clearly separated, in that order
3. Sphere power with sign
4. Cylinder power with sign and axis in three-digit notation
5. Add power, and the segment type or height if relevant
6. Prism power and base direction if prescribed
7. Interpupillary distance, distance and near where applicable
8. Vertex distance for powers over ±4.00 D
9. Any special instructions: lens material, tint, coatings, occupational use
10. Expiry date where required by jurisdiction
11. Prescriber's name, signature, licence number and contact details

A prescription missing the axis, the base direction of a prism, or the interpupillary distance cannot be dispensed safely and must be clarified, not guessed.