Difference between revisions of Diopters

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(→‎Cylinder: change variable to P for power, instead of F for focal power, since f is used for focal length)
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=== Cylinder ===
=== Cylinder ===
A cylindrical lens of power F<sub>c</sub> has focal power F at angle θ from its axis:
A cylindrical lens of focal power P<sub>cyl</sub> has power P at angle θ from its axis:


<math>F = F_{cyl} (\sin\theta)^2</math>
<math>P = P_{cyl} (\sin\theta)^2</math>
 
==== Axis ====
Axis is usually in degrees modulo 180. It is popular for 0 to be written as 180 in some areas.


==== Transposition ====
==== Transposition ====
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<math>F = F_{cyl} \left( 1 - (\cos\theta)^2 \right) = F_{cyl} + \left(-F_{cyl}\right)(\sin{\left(\theta - 90^{\circ}\right)})^2</math>
<math>P = P_{cyl} \left( 1 - (\cos\theta)^2 \right) = P_{cyl} + \left(-P_{cyl}\right)(\sin{\left(\theta - 90^{\circ}\right)})^2</math>


We can see that by adding the cylindrical power to the spherical power and inverting the cylinder and adding 90 degrees (same as subtracting 90 degrees, since the axis is in modulo 180 degrees) to the axis, we get an equivalent combination.
We can see that by adding the cylindrical power to the spherical power and inverting the cylinder and adding 90 degrees (same as subtracting 90 degrees, since the axis is in modulo 180 degrees) to the axis, we get an equivalent combination.
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By calculating the average value over all angles using an integral, the result<ref>it is only necessary to integrate one or two periods: https://www.wolframalpha.com/input/?i=average+of+%28sin+x%29%5E2+from+0+to+2+pi</ref> is
By calculating the average value over all angles using an integral, the result<ref>it is only necessary to integrate one or two periods: https://www.wolframalpha.com/input/?i=average+of+%28sin+x%29%5E2+from+0+to+2+pi</ref> is


<math>F_{avg} = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} F_{cyl} (\sin{t})^2 \,dt = \frac{1}{2} F_{cyl}</math>
<math>P_{avg} = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} P_{cyl} (\sin{t})^2 \,dt = \frac{1}{2} P_{cyl}</math>


This is why the spherical equivalent has power equal to half of the cylinder's power.
This is why the spherical equivalent has power equal to half of the cylinder's power.
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<math>F = F_{cyl} (\sin{\left(\theta + \phi\right)})^2 = F_{cyl} \left( \frac{1-\cos{\left(2\theta + 2\phi\right)}}{2} \right) = \frac{1}{2} F_{cyl} + \frac{-F_{cyl}}{2} \cos{\left(2\theta + 2\phi\right)}</math>
<math>P = P_{cyl} (\sin{\left(\theta + \phi\right)})^2 = P_{cyl} \left( \frac{1-\cos{\left(2\theta + 2\phi\right)}}{2} \right) = \frac{1}{2} P_{cyl} + \frac{-P_{cyl}}{2} \cos{\left(2\theta + 2\phi\right)}</math>


The constant parts are added with the spherical components. The cosines can be added by converting them to [https://en.wikipedia.org/wiki/Phasor phasors] and adding the phasors together. The resulting phasor corresponds to one of two cylindrical lenses (see the section on Transposition), and its corresponding spherical component must be subtracted from the total spherical component.
The constant parts are added with the spherical components. The cosines can be added by converting them to [https://en.wikipedia.org/wiki/Phasor phasors] and adding the phasors together. The resulting phasor corresponds to one of two cylindrical lenses (see the section on Transposition), and its corresponding spherical component must be subtracted from the total spherical component.
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