Formula cheat-sheets

f (m) = 1 / F (D)        F = 1 / f

Example: A +4.00 D lens → f = 0.25 m (25 cm)

Convert centimetres to metres before dividing: 25 cm = 0.25 m.

nominal power = front (base) curve + back curve

Example: Front curve +6.00 D, back curve −8.00 D → −2.00 D

F = (n₂ − n₁) / r

Where n₁ = index light leaves, n₂ = index it enters, r = radius in metres.

Example: Air to n = 1.50, radius 0.10 m → +5.00 D

Awaiting SME confirmation against a primary source.

true power = clock reading × (n − 1) / (1.53 − 1)

Where n = index of the lens being measured; the clock is calibrated for 1.53.

Example: Clock reads +5.00 D on a 1.586 lens → +5.53 D

Awaiting SME confirmation against a primary source.

new sphere = S + C    new cyl = −C    new axis = axis ± 90 (keep 1–180)

Example: +1.00 −2.50 × 090 → −1.50 +2.50 × 180

Add, change the sign, change the axis. An axis of 0 is written 180.

SE = S + C / 2

Example: −2.00 −1.50 × 180 → −2.75 D

F = S + C × {0, 25, 50, 75, 100 %} for 0, 30, 45, 60, 90° away from the axis

Example: −1.00 −2.00 × 030, power in the 180 meridian (30° off axis) → −1.50 D

On the axis the cylinder adds nothing; 90° away it adds all of its power.

F(θ) = S + C × sin²(θ − axis)

Example: −1.00 −2.00 × 020, power in the 180 meridian → −1.23 D

Gives the table values at 0/30/45/60/90° and works at any angle.

Δ = F × d(mm) / 10        (or Δ = F × c, with c in cm)

Where F = power in the meridian of decentration, d = distance of the OC from the pupil.

Example: +4.00 D lens, OC 3 mm out from the pupil → 1.20Δ BO

Plus lens: base goes the same way the OC moved. Minus lens: the opposite way.

d (mm) = 10 × Δ / F

Example: 2Δ BI wanted from a −5.00 D lens → 4 mm, OC moved out

P = −F · r    F = [[S + C sin²α, −C sinα cosα], [−C sinα cosα, S + C cos²α]]

Where α = cylinder axis, r = position of the pupil from the OC in cm.

Example: OD +2.00 −1.00 × 045, pupil 4 mm nasal and 2 mm below the OC → 0.50Δ BO and 0.10Δ BU

An oblique cylinder adds vertical prism to a horizontal decentration; the exam table does not.

Awaiting SME confirmation against a primary source.

P = √(H² + V²)    θ = atan(V / H)    H = P cos θ    V = P sin θ

Where 360° notation: 0° at the examiner’s right, 90° = base up.

Example: OD 4Δ BI and 3Δ BU → 5.00Δ at 36.9°

Awaiting SME confirmation against a primary source.

imbalance = |Δ OD − Δ OS|, each Δ = vertical power × reading depth (mm) / 10

Example: OD −1.00 D, OS −3.50 D, reading 10 mm below the OCs → 2.50Δ

Slab-off is commonly considered from about 1.5Δ (awaiting SME confirmation).

Awaiting SME confirmation against a primary source.

DC = mm × F² / 1000

Example: +10.00 D lens moved 3 mm → +0.30 D

moved out (away from the eye): EP = F + DC    moved in: EP = F − DC

Example: +10.00 D refracted at 12 mm, worn at 15 mm → +10.30 D

Effective power is what the patient gets from the unchanged lens in its new place.

moved out: CP = F − DC    moved in: CP = F + DC

Example: +10.00 D refracted at 12 mm, worn at 15 mm → +9.70 D

Compensated power is what to order so the patient still gets the refracted power.

F new = F / (1 − d × F)

Where d in metres, positive when the lens moves toward the eye.

Example: −10.00 D at 12 mm, moved to the eye → −8.93 D

compensate each principal meridian, round each, rebuild in minus cylinder

Example: −8.00 −2.00 × 180 at 12 mm, moved to the eye → −7.25 −1.50 × 180

frame PD = A + DBL

Example: A 52 mm, DBL 18 mm → 70 mm

decentration = (frame PD − patient PD) / 2

Example: Frame PD 70 mm, patient PD 64 mm → 3 mm in, each lens

Divide by two: each lens moves half of the difference.

OC offset per lens = (fitted PD − patient PD) / 2

Example: Lenses made at 68 mm for a 62 mm PD → 3 mm out, each lens

Monocular OC heights are not halved.

MBS = ED + total decentration (= ED + 2 × per-lens)

Example: ED 56 mm, frame PD 70 mm, patient PD 64 mm → 62 mm

Many labs add about 2 mm for edging (awaiting SME confirmation).

position = height − B / 2    (+ above, − below the datum line)

Example: Seg height 20 mm, B 44 mm → 2 mm below the datum line

n = c / v

Where c = speed of light in a vacuum, v = speed in the material.

Example: Light travels at 199,861,639 m/s in the material → n = 1.5

TCA = Δ / Abbe value

Example: 4Δ at the viewing point, Abbe value 30 → 0.13Δ

Awaiting SME confirmation against a primary source.

R = ((n − 1) / (n + 1))²

Example: Uncoated n = 1.50 surface → 4 %

Awaiting SME confirmation against a primary source.