What Is Resultant Prism?
When two or more prisms act on the same eye, their combined effect is called the resultant prism. Because prisms have both magnitude (power in prism diopters) and direction (base orientation), they behave as vectors. You cannot simply add the numbers together unless the prisms share the same base direction.
Same-Direction Prisms
When two prisms have the same base direction, just add their powers:
3Δ base up + 2Δ base up = 5Δ base up
When two prisms have opposite base directions, subtract:
5Δ base out - 2Δ base in = 3Δ base out
Prisms at Right Angles
The most common ABO exam scenario involves combining a horizontal prism (base in or base out) with a vertical prism (base up or base down). Use the Pythagorean theorem:
Resultant = √(H² + V²)
The direction is found using trigonometry:
tan(θ) = V / H
where θ is measured from the horizontal.
Worked Example
Combine 3Δ base out with 4Δ base up (right eye):
Resultant = √(3² + 4²) = √(9 + 16) = √25 = 5Δ
θ = arctan(4/3) = 53.1°
The resultant is 5Δ at 53° from the horizontal, in the base-out-and-up direction. In TABO notation for the right eye, this would be approximately 5Δ at 127° (since base out for the right eye is at 180° and base up is at 90°, the resultant direction is in the second quadrant).
The Graphical Method
You can also solve prism combination problems graphically:
- Draw the first prism as a vector arrow with length proportional to its power
- From the tip of the first arrow, draw the second prism vector
- The resultant is the arrow from the starting point to the final tip
- Measure the length (power) and angle (direction) of the resultant
This tip-to-tail method works for any number of prisms at any angle, not just those at 90°.
Resolving Prism Into Components
The reverse process is also tested: given a single oblique prism, break it into horizontal and vertical components.
For a prism of power P at angle θ from horizontal:
- Horizontal component = P × cos(θ)
- Vertical component = P × sin(θ)
Compounding and Resolving on the ABO Exam
The ABO exam may ask you to:
- Find the resultant of two prisms at 90° (use Pythagorean theorem)
- Resolve an oblique prism into horizontal and vertical components (use sin/cos)
- Determine the net vertical or horizontal prism when both eyes have prescribed prism
Fresnel Prism and Large Resultants
When a large resultant prism is needed (over 5-6Δ), ground-in prism becomes heavy and causes noticeable distortion. In these cases, Fresnel press-on prisms offer a temporary or trial solution. The resultant calculation is the same regardless of whether the prism is ground-in or Fresnel.
Key Takeaways
- Prisms are vectors with both magnitude and direction
- Same-direction prisms add; opposite-direction prisms subtract
- For perpendicular prisms, use the Pythagorean theorem: R = √(H² + V²)
- Find the resultant angle with tan(θ) = V / H
- Resolve oblique prisms using cos (horizontal) and sin (vertical)