Vertex Calculation Oddities

I received the following email from Eli today:

I have a question about your vertex calculator. If I try to vertex correct a refraction of -9.00-2.00x180, the diagram of the power cross that comes up on your website shows -8.00 in one meridian and -9.75 in the other meridian (a cylinder amount of -1.75), but the vertexed prescription is listed with -1.50 cylinder. It just seems like there is a discrepancy between your diagram and the vertexed prescription written just below it. Thanks for your help in understanding this.

vertexted refraction on optical crosses

This question pops up every once and a while, and it’s a good one.  This looks like a mistake, or at least a discrepancy, but it’s really simply a phenomena of rounding.

If you take the -11.00 D meridian and vertex it, the actual value you get is -9.72. However, I like the optical cross labels to look nice and tidy, so I wrote the program to round this up to the nearest quarter diopter, -9.75.

Similarly, the -9.00 D meridian ends up vertexing to -8.12. This gets rounded to -8.00 (but just barely - if it was more than -8.125 it would have been rounded to -8.25).

As a consequence, one meridian was rounded up to -9.75 and the other rounded down to -8.00, so the optical cross seems to show a -1.75 D cyl power.

However, the suggested cyl power uses the true calculated values, -9.72 and -8.12, and subtracts them to get the cyl power, which turns out to be -1.60 D in this case. Because I still wanted things to look pretty - and to be easier to compare to our real world contact lens options - this is rounded to 1.50 D.

In summary, when you see a discrepancy between the cylinder on the optical cross and the cylinder on the suggested Rx, the true, exact number rests somewhere between these values. However, rest assured that the cylinder in the suggested Rx will always be within an 1/8 D of the exact calculated power.

I hope that makes sense!