Difference between revisions of Diopters
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==== Adding Lenses ==== | ==== Adding Lenses ==== | ||
Multiple lens, each with spherical and cylindrical components (not necessarily at the same axis) can be added to form one lens with a spherical and cylindrical component. | |||
We can use the double angle formula to convert each cylindrical lens into a constant plus a cosine: | We can use the double angle formula to convert each cylindrical lens into a constant plus a cosine: | ||
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<math>F = F_{cyl} (\sin\theta)^2 = F_{cyl} \left( \frac{1-\cos{2\theta}}{2} \right) = \frac{1}{2} F_{cyl} + \frac{-F_{cyl}}{2} \cos{2\theta}</math> | <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> | ||
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 equivalent 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 equivalent must be subtracted from the total spherical component. | ||