Difference between revisions of Diopters
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====Examples==== | ====Examples==== | ||
"Full correction" takes an object at infinity and produces a virtual image at your far point distance d: | "Full correction" takes an object at infinity and produces a virtual image at your far point distance d: | ||
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This is the resulting equation at the beginning of the article. It also explains why the focal power is increased for objects at closer distances: mainstream optometry calls this the "add" for presbyopia. For example, if you choose 80 cm as the working distance for your [[differentials]] (resulting in a +1.25 dpt "add"), and your blur horizon is 50 cm (resulting in -2 dpt), the formula is | This is the resulting equation at the beginning of the article. It also explains why the focal power is increased for objects at closer distances: mainstream optometry calls this the "add" for presbyopia, although they typically use the minimum amount required for you to see at 40 cm with full distance correction using accommodation. For example, if you choose 80 cm as the working distance for your [[differentials]] (resulting in a +1.25 dpt "add"), and your blur horizon is 50 cm (resulting in -2 dpt), the formula is | ||
<math>\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{80\ cm}+\frac{1}{-50\ cm}=1.25\ dpt + \left(-2\ dpt\right) = -0.75\ dpt</math> | <math>\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{80\ cm}+\frac{1}{-50\ cm}=1.25\ dpt + \left(-2\ dpt\right) = -0.75\ dpt</math> | ||
=== 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 power F<sub>c</sub> has focal power F at angle θ from its axis: | ||
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==== Transposition ==== | ==== Transposition ==== | ||
We can understand why there are two different ways to write | We can understand why there are two different ways to write a spherical and cylindrical lens combination, using the Pythagorean trigonometric identity and the complementary angle identity: | ||
<math>(\sin\theta)^2 + (\cos\theta)^2 = 1</math> | <math>(\sin\theta)^2 + (\cos\theta)^2 = 1</math> | ||
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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>F = F_{cyl} \left( 1 - (\cos\theta)^2 \right) = F_{cyl} + \left(-F_{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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For example, -1 sph -1 cyl 1 axis is the same as -2 sph +1 cyl 91 axis. | For example, -1 sph -1 cyl 1 axis is the same as -2 sph +1 cyl 91 axis. | ||
In general optometrists prefer to use a negative cylinder and ophthalmologists prefer to use a positive cylinder, but the two forms are equivalent. | In general, optometrists prefer to use a negative cylinder and ophthalmologists prefer to use a positive cylinder, but the two forms are equivalent. | ||
==== Spherical Equivalent ==== | ==== Spherical Equivalent ==== | ||
By calculating the average value over all angles using an integral, the | By calculating the average value over all angles using an integral, the result is | ||
<math>F_{avg} = \frac{1}{2} F_{cyl}</math> | <math>F_{avg} = \frac{1}{2} F_{cyl}</math> | ||
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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\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> | ||
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, 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. | ||
==References== | ==References== | ||
{{reflist}} | {{reflist}} | ||
[[Category:Articles]] | [[Category:Articles]] | ||