Difference between revisions of Diopters

Line 60:

This is also sometimes presented in the Newtonian form:

<math>\left(d_o-f\right)\left(d_i-f\right)=f^2</math>

====Examples====

Line 72:

Line 71:

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

<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 ===

A cylindrical lens of power F<sub>c</sub> has focal power F at angle θ from its axis:

<math>F = F_{cyl} (\sin\theta)^2</math>

==== Transposition ====

We can understand why there are two different ways to write the same spherical and cylindrical lens components, using the Pythagorean trigonometric identity and the complementary angle identity:

<math>(\sin\theta)^2 + (\cos\theta)^2 = 1</math>

<math>\cos\theta = \sin{\left(90^{\circ} - \theta\right)} = -\sin{\left(\theta - 90^{\circ}\right)}</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.

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.

==== Spherical Equivalent ====

By calculating the average value over all angles using an integral, the following result is

This is why the spherical equivalent has power equal to half of the cylinder's power.

==== Adding Lenses ====

Two 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:

<math>\cos{2\theta}=1-2(\sin\theta)^2</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.

==References==

[[Category:Articles]]

@@ Line 60: / Line 60: @@
 This is also sometimes presented in the Newtonian form:
-<math>(d_o-f)(d_i-f)=f^2</math>
+<math>\left(d_o-f\right)\left(d_i-f\right)=f^2</math>
 ====Examples====
@@ Line 72: / Line 71: @@
 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
-<math>\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{80\ cm}+\frac{1}{-50\ cm}=1.25\ dpt + (-2\ dpt) = -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 ===
+A cylindrical lens of power F<sub>c</sub> has focal power F at angle θ from its axis:
+<math>F = F_{cyl} (\sin\theta)^2</math>
+==== Transposition ====
+We can understand why there are two different ways to write the same spherical and cylindrical lens components, using the Pythagorean trigonometric identity and the complementary angle identity:
+<math>(\sin\theta)^2 + (\cos\theta)^2 = 1</math>
+<math>\cos\theta = \sin{\left(90^{\circ} - \theta\right)} = -\sin{\left(\theta - 90^{\circ}\right)}</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.
+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.
+==== Spherical Equivalent ====
+By calculating the average value over all angles using an integral, the following result is
+<math>F_{avg} = \frac{1}{2} F_{cyl}</math>
+This is why the spherical equivalent has power equal to half of the cylinder's power.
+==== Adding Lenses ====
+Two 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:
+<math>\cos{2\theta}=1-2(\sin\theta)^2</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.
 ==References==
 {{reflist}}
 [[Category:Articles]]

Difference between revisions of Diopters

Navigation menu

Search