Prism Diopter Calculator (Prentice’s Rule & Deviation)

How it works

Prentice: Δ = F (D) × decentration (cm) · plus lens: base toward the optical centre; minus lens: base away from it · spherocylinder: horizontal power = S + C sin²(axis), vertical power = S + C cos²(axis) (full power matrix used for oblique axes) · net horizontal = OD + OS (base in +); vertical imbalance = OD − OS (base up +) · θ = arctan(Δ ÷ 100) · resultant = √(H² + V²)

A spectacle lens acts like a stack of prisms whose power grows away from the optical centre, so looking through any other point bends the light. Prentice’s rule gives the amount: lens power in dioptres times the distance from the optical centre in centimetres. A plus lens is thickest in the middle, so its base points toward the optical centre; a minus lens is thinnest there, so its base points away. Opticians use this on purpose to create prescribed prism by moving the optical centre, and it explains why glasses made to the wrong PD, or bifocal wearers with unequal prescriptions reading through the lower lens, get eyestrain or double vision. The both-eyes modes take sphere, cylinder and axis for each eye: the power acting horizontally is S + C·sin²(axis) and vertically S + C·cos²(axis), and for oblique axes the calculator uses the full dioptric power matrix, which also captures the small cross-meridian prism. Horizontal prism adds between the eyes when the bases point the same way (both base in) and cancels when they differ; vertical prism matters as the difference between the eyes. ANSI Z80.1 allows about 0.67 Δ of horizontal and 0.33 Δ of vertical imbalance. A prism dioptre is a displacement (1 cm at 1 m), so converting to degrees uses a tangent, and horizontal and vertical prism combine like vectors into one oblique prism written in 360° notation.

Worked example

A −6.00 D lens with its optical centre 4 mm out from the pupil: 6 × 0.4 = 2.40 Δ base in, well over the 0.67 Δ horizontal tolerance. Both eyes: OD −4.00 −1.00 × 90 and OS −2.00 made at a PD of 66 for a 62 mm PD put each centre 2 mm out; the right lens acts as −5.00 D horizontally (1.00 Δ base in) and the left as −2.00 D (0.40 Δ base in), a net 1.40 Δ base in. Reading 10 mm below the centres with OD +1.00 and OS +3.00 gives 1.00 Δ and 3.00 Δ base up, a 2.00 Δ vertical imbalance. And 3 Δ base in with 2 Δ base up on the right eye resolves to 3.61 Δ at 34°.

Sources: ANSI Z80.1 Prescription Ophthalmic Lenses — Recommendations (The Vision Council) · Laramy K. Prism — lens form and theory (Prentice’s rule, compounding and resolving prism) · OpticianWorks — Ultimate ABO Study Guide (Prentice’s rule, induced and vertical imbalance)

Frequently asked questions