Difference between revisions of Optics related math

(7 intermediate revisions by 2 users not shown)

Line 1:

[[File:Math guy with glasses.gif|right]]

Here's a page with maths related to diopters and glasses.

Line 33:

A diverging lens behaves in a similar way, but for distant objects - the light rays incident from the source object are refracted outwards so that they are diverging even faster. Again, a virtual focus is said to form behind the lens. The thin lens equation still works as long as you use negative numbers to describe both the (virtual) image location and the focal length.

The corrective lens for [[myopia]] is a diverging lens. It works by forming a virtual image of distant objects, and it is this virtual image that the near-sighted eye is able to focus on.

The corrective lens for [[myopia]] is a diverging lens. It works by forming a virtual image of distant objects, and it is this virtual image that the near-sighted eye is able to focus on. (You can also choose to think of it as lens in series with your eye, forming a compound lens with lower power and therefore a longer focal length.)

===Diopters are inverse meters===

Line 45:

Remember that 100cm = 1m.

<math>D = \frac{1}{meters}</math>

<math>D = \frac{1}{ \text{meters} }</math>

conversely

<math>meters = \frac{1}{D}</math>

<math> \text{meters} = \frac{1}{D}</math>

==Calculating correction==

Line 94:

==Visual acuity ~~equation~~==

==Visual acuity==

<math>(\frac{~~font~~\ ~~height~~}{~~distance~~\ to\ ~~sign~~})(\frac{180}{pi}) \times ~~60 = arcminutes~~ = a</math>

[[File:TrigonometryTriangle.svg|right|TrigonometryTriangle]]

A reminder about some trigonometry / geometry

# <math>\tan(A) = \frac{a}{b}</math>

#* so for fixed angle, 'a' is proportional to 'b'

# for very small angles (expressed in radians), <math>\tan(A) \approx A</math>

#* so at small angles, 'a' is proportional to 'A'

# <math>1 \text{ radian} = \frac{180}{\pi} \text{ degrees} = \frac{60 \times 180}{\pi} \text{ arc-minutes} </math>

Acuity is a measure of the ability to resolve small details, defined by the "minimum angle of resolution". Normal vision ("20/20", or 1.0) is the ability to resolve features subtending 1 minute of arc. In order to have a bigger-is-better score, the reciprocal of the angle is usually used:

<math> \text{Acuity} = 1/A = \frac{pi}{60 \times 180} \times \frac{b}{a} = 0.00029 \times \frac{b}{a}</math>

An alternative representation of acuity is "logMAR" ("log<sub>10</sub> Minimum Angle of Resolution"). This has a bigger-is-worse direction.

* 0.0 corresponds to "20/20" (1.0 or 1 arc-minute)

* 1.0 is "20/200" (0.1 or 10 arc-minutes)

* -0.3 is "20/10" (2.0 or 0.5 arc-minutes)

~~Note~~: ~~5Arcminutes = 20/20~~

On a [[Snellen chart]] the letters are defined on a 5x5 grid: the detail to be resolved to distinguish between characters is one pixel of that grid.

Here 'b' is the distance to the chart (eg 6m) and 'a' is the height of the letters. It's not critical that the eye is at the bottom or top of a row of letters - as long as the angles remain small, the numerical error is insignificant.

~~Set up proportion: <math>\frac{a}{(\frac{20}{x})} = \frac{5}{(\frac{20}{~~20~~})}<~~/~~math>~~

Acuity is reported in the form "distance/letter-row", such as "20/40", etc.

~~===Visual acuity (mm/metres)===~~

<math> \text{Acuity} = 5 \times 0.00029 \times \frac{ \text{distance to chart} }{ \text{letter height} }

<math>\frac{~~font~~\ height ~~(mm)~~}{distance\ to~~\ sign(m)~~} \times ~~13.75 = denominator~~ \~~times of \frac~~{20}{x}</~~math~~>

= \frac{ \text{distance to chart} }{690 \times \text{letter height} } </math>

<ref>{{cite jake|https://endmyopia.org/use-math-to-turn-any-text-into-your-own-impromptu-eyechart/}}</ref>

~~===Visual acuity (in~~/~~feet)===~~

Then, for example

<math>\~~frac~~{~~font\~~ height~~(in)~~}~~{distance~~\ to\ ~~sign(ft~~.)} \times ~~1146~~ = ~~denominator~~ \~~times of~~ \~~frac~~{20}{x}</math>

* for the "6/12" line, we need <math>690 \times \text{height} = 12m \implies \text{height} = 12m / 690 = 1.73cm</math>

* for the "20/15" line, we need <math>690 \times \text{height} = 15ft \implies \text{height} = 15 * 12 / 690 = 0.26 \text{inch}</math>

~~With text that we are familiar with,~~ the ~~brain may clear up that text more than our vision would actually allow.<ref>{{cite jake|https://endmyopia~~.~~org/use-math-to-turn-any-text-into-your-own-impromptu-eyechart/}}</ref>~~

The denominator can be interpreted as the distance at which the letters subtend 5 minutes of arc.

==Average axial length accomodation/rate of change==

~~<math>~~typical\ emmetropic\ eye = 25mm ~~= 25,000\ microns</math>~~

* The typical emmetropic eye is 25mm

* change in axial length of 1mm = 3D

~~<math>~~change\ in\ axial\ length\ of\ 1mm=3D~~</math>~~

If someone with typical eyes wanted to adapt say 20/20 to .25 less

Line 119:

Line 137:

about 0.92microns/day - 0.69microns/day average

''Credit: [https://www.facebook.com/groups/endmyopia/permalink/759426960917375/ Mark Podowski]''

==Converting from Glasses to Contact Lens Prescription or vice-versa==

* [[Vertex distance]]

* [[Wikipedia:Vertex_distance|Vertex distance formula (also for astigmatism)]]

==See Also==

[https://wiki.endmyopia.org/wiki/Guide:Reducing_differentials#First_differentials Guide:Reducing differentials > First differentials]

==References==

@@ Line 1: / Line 1: @@
-[[File:Math guy with glasses.gif|right]]
+{{gif fixer|[[File:Math guy with glasses.gif]]|right}}
 Here's a page with maths related to diopters and glasses.
@@ Line 33: / Line 33: @@
 A diverging lens behaves in a similar way, but for distant objects - the light rays incident from the source object are refracted outwards so that they are diverging even faster. Again, a virtual focus is said to form behind the lens. The thin lens equation still works as long as you use negative numbers to describe both the (virtual) image location and the focal length.
-The corrective lens for [[myopia]] is a diverging lens. It works by forming a virtual image of distant objects, and it is this virtual image that the near-sighted eye is able to focus on.
+The corrective lens for [[myopia]] is a diverging lens. It works by forming a virtual image of distant objects, and it is this virtual image that the near-sighted eye is able to focus on. (You can also choose to think of it as lens in series with your eye, forming a compound lens with lower power and therefore a longer focal length.)
 ===Diopters are inverse meters===
@@ Line 45: / Line 45: @@
 Remember that 100cm = 1m.
-<math>D = \frac{1}{meters}</math>
+<math>D = \frac{1}{ \text{meters} }</math>
 conversely
-<math>meters = \frac{1}{D}</math>
+<math> \text{meters} = \frac{1}{D}</math>
 ==Calculating correction==
@@ Line 94: / Line 94: @@
-==Visual acuity equation==
+==Visual acuity==
-<math>(\frac{font\ height}{distance\ to\ sign})(\frac{180}{pi}) \times 60 = arcminutes = a</math>
+[[File:TrigonometryTriangle.svg|right|TrigonometryTriangle]]
+A reminder about some trigonometry / geometry
+# <math>\tan(A) = \frac{a}{b}</math>
+#* so for fixed angle, 'a' is proportional to 'b'
+# for very small angles (expressed in radians), <math>\tan(A) \approx A</math>
+#* so at small angles, 'a' is proportional to 'A'
+# <math>1 \text{ radian} = \frac{180}{\pi} \text{ degrees} = \frac{60 \times 180}{\pi} \text{ arc-minutes} </math>
+Acuity is a measure of the ability to resolve small details, defined by the "minimum angle of resolution". Normal vision ("20/20", or 1.0) is the ability to resolve features subtending 1 minute of arc. In order to have a bigger-is-better score, the reciprocal of the angle is usually used:
+<math> \text{Acuity} = 1/A = \frac{pi}{60 \times 180} \times \frac{b}{a} = 0.00029 \times \frac{b}{a}</math>
+An alternative representation of acuity is "logMAR" ("log<sub>10</sub> Minimum Angle of Resolution"). This has a bigger-is-worse direction.
+* 0.0 corresponds to "20/20" (1.0 or 1 arc-minute)
+* 1.0 is "20/200" (0.1 or 10 arc-minutes)
+* -0.3 is "20/10" (2.0 or 0.5 arc-minutes)
-Note: 5Arcminutes = 20/20
+On a [[Snellen chart]] the letters are defined on a 5x5 grid: the detail to be resolved to distinguish between characters is one pixel of that grid.
+Here 'b' is the distance to the chart (eg 6m) and 'a' is the height of the letters. It's not critical that the eye is at the bottom or top of a row of letters - as long as the angles remain small, the numerical error is insignificant.
-Set up proportion: <math>\frac{a}{(\frac{20}{x})} = \frac{5}{(\frac{20}{20})}</math>
+Acuity is reported in the form "distance/letter-row", such as "20/40", etc.
-===Visual acuity (mm/metres)===
+<math> \text{Acuity} = 5 \times 0.00029 \times \frac{ \text{distance to chart} }{ \text{letter height} }
-<math>\frac{font\ height (mm)}{distance\ to\ sign(m)} \times 13.75 = denominator \times of \frac{20}{x}</math>
+= \frac{ \text{distance to chart} }{690 \times \text{letter height} } </math>
+<ref>{{cite jake|https://endmyopia.org/use-math-to-turn-any-text-into-your-own-impromptu-eyechart/}}</ref>
-===Visual acuity (in/feet)===
+Then, for example
-<math>\frac{font\ height(in)}{distance\ to\ sign(ft.)} \times 1146 = denominator \times of \frac{20}{x}</math>
+* for the "6/12" line, we need <math>690 \times \text{height} = 12m \implies \text{height} = 12m / 690 = 1.73cm</math>
+* for the "20/15" line, we need <math>690 \times \text{height} = 15ft \implies \text{height} = 15 * 12 / 690 = 0.26 \text{inch}</math>
-With text that we are familiar with, the brain may clear up that text more than our vision would actually allow.<ref>{{cite jake|https://endmyopia.org/use-math-to-turn-any-text-into-your-own-impromptu-eyechart/}}</ref>
+The denominator can be interpreted as the distance at which the letters subtend 5 minutes of arc.
 ==Average axial length accomodation/rate of change==
-<math>typical\ emmetropic\ eye = 25mm = 25,000\ microns</math>
+* The typical emmetropic eye is 25mm
+* change in axial length of 1mm = 3D
-<math>change\ in\ axial\ length\ of\ 1mm=3D</math>
 If someone with typical eyes wanted to adapt say 20/20 to .25 less
@@ Line 119: / Line 137: @@
 about 0.92microns/day - 0.69microns/day average
 ''Credit: [https://www.facebook.com/groups/endmyopia/permalink/759426960917375/ Mark Podowski]''
 ==Converting from Glasses to Contact Lens Prescription or vice-versa==
 * [[Vertex distance]]
 * [[Wikipedia:Vertex_distance|Vertex distance formula (also for astigmatism)]]
+==See Also==
+[https://wiki.endmyopia.org/wiki/Guide:Reducing_differentials#First_differentials Guide:Reducing differentials > First differentials]
 ==References==
 {{reflist}}