The formulas
R = √(H² + V²)
θ = arctan(V ÷ H)
- H, V
- Horizontal and vertical prism in prism dioptres.
- R
- The single resultant prism.
- θ
- Base angle measured from the horizontal, 0° to 90°. The calculator gives this angle; you place it in the right quadrant.
To split a prism back into its parts (resolving), work backwards by hand: H = R × cos θ and V = R × sin θ. For example, 5 Δ at 30° from the horizontal is 4.33 Δ horizontal and 2.50 Δ vertical.
Worked example
Right eye: 4.00 Δ base out and 3.00 Δ base up. With H = 4.00 and V = 3.00 the calculator shows 5.00 Δ and Base Angle: 36.9°. Base up and out on a right eye is the 90–180° quadrant of the 360° chart, so the order reads 5.00 Δ at 180 − 36.9 = 143°. The same prism on a left eye would be at 37°.
Placing the angle on the 360° chart
The 360° notation is read as the practitioner faces the wearer, with 0° to the practitioner's right for both eyes. For the right eye 0° is base in; for the left eye 0° is base out. Up is 90° and down is 270° for both.
| Base direction | Right eye (OD) | Left eye (OS) |
|---|---|---|
| Up and in | θ | 180° − θ |
| Up and out | 180° − θ | θ |
| Down and in | 360° − θ | 180° + θ |
| Down and out | 180° + θ | 360° − θ |
When it is used
- Writing a lab order when a prescription has prism in both directions: a lens can carry only one prism, so the two are combined.
- Checking a finished lens: on a lensmeter the target moves diagonally, by the resultant, not by two separate amounts.
- Splitting prism between the eyes to even out thickness, after resolving it into its horizontal and vertical parts.
Limits. The calculator takes magnitudes only, from 0 to 15 Δ in 0.25 Δ steps, and does not know the eye or the base directions. It adds prisms as vectors, which is accurate for the small amounts used in spectacles. See the medical disclaimer.
Questions
Why can't the lab grind the two prisms separately?
A lens has one front and one back surface, so it can be tilted only one way. A horizontal and a vertical prism together make one tilt with one base direction: the resultant.