Sorry for the delayed response; I've been traveling.
Yes, I should have been clearer about that point.
Fortunately, the problem can be simplified considerably, since the cylinder powers are equal in magnitude and opposite in sign. In this case, the equation for the resultant cylinder power simplifies to:
where Ang is the angle between the two cylinder axes. This is also the equation for a
Stokes lens. The sign of this resultant cylinder can be set to the desired cylinder convention (either plus or minus).
The new sphere power is given simply by:
The new cylinder axis is simply midway between the original cylinder axes, once both have been transposed to the same cylinder convention.
In our example, we have:
Plano +3.00 x 180 (eye)
Plano -3.00 x 020 (lens)
The original angle between these two cylinders is 20 degrees. Using our Stokes lens equation, we have for the resultant cylinder power:
And the resultant sphere power is:
Finally, after transposing the two original cylinder powers into minus cylinder form (i.e., +3.00 -3.00 x 090 and Plano -3.00 x 020), we arrive at the new cylinder axis midway between the two: (90 + 20) / 2 = 55.
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