Difference between revisions of Optics related math

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

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

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__TOC__

==~~The thin lens equation~~==

==Lenses==

[[File:Lens3.svg|right]]

The image shows a typical converging lens. The three rays drawn each have an interesting characteristic:

* the top ray enters the lens parallel with the optic axis, and so passes through the focal point on the other side

* the middle ray passes through the optical centre of the lens, and is undeviated

* the bottom ray passes through the focal point on the incident side, and so emerges parallel to the optic axis

===The thin lens equation===

where

* <math>f</math> = focal length of lens

* <math>s</math> = distance to object

* <math>s</math> = distance to object (<math>S_1</math> on the picture)

* <math>s'</math> = distance to image

* <math>s'</math> = distance to image (<math>S_2</math> on the picture)

===Infinity===

The term '''object at infinity''' is often used. When <math>s=\infty</math> is substituted into the thin lens equation, that term vanishes, so that the focal length is then just the image location. For any sufficiently large object distance, the contribution from the reciprocal becomes negligible.

As the object approaches the lens, the outgoing rays converge less and less, until the object reaches the focus of the lens (<math>s=f</math>), and the transmitted light is parallel - we say it "focuses at <math>s'=\infty</math>", which means it never comes together into a focus. (This can also be described as a virtual image at <math>s'=-\infty</math> - see below.)

===Virtual image===

~~A converging lens (such as a magnifying glass) behaves like a "typical" lens - the incoming light is brought to a focus on the opposite side of the lens.~~

A diverging lens behaves ~~differently~~ - the light rays ~~spreading~~ from the source object are refracted outwards so that they are diverging even faster. ~~They are not brought to~~ a focus ~~in any intuitive sense. Instead, the light behaves ''as if'' it was coming from a closer object. This~~ is ~~termed a '''virtual''' image - it lies between the source object and~~ 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.

When the object is brought even closer to the lens (<math>s < f</math>) the emerging rays are now diverging. When substituted into the equation, <math>s' < 0</math>. This is interpreted as a '''virtual image''', ''behind'' the lens. This is the mode in which reading glasses (plus lenses) are used - the eye is able to focus on the virtual image which appears to be further away than the real source object.

===Diverging lens===

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. (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.)

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

===Diopters are inverse meters===

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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==

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But rather than actually placing such a lens in front of your full prescription,

simply ~~subtract~~ that value.

simply add that value (being careful with signs).

ie if your screen is 50cm away, that corresponds to a power of <math>\frac{1}{0.5} = 2D</math>.

So that's the value you'd ~~subtract from~~ your full prescription.

So that's the value you'd add to your full prescription (resulting in a less-negative lens).

===From blur horizon of naked eye===

~~From your~~ cm measurement...

Note that this does not take either [[cylinder]] ([[astigmatism]]) or [[vertex distance]] into account.

Your [[cm measurement]] gives you the distance the eye can see when it is fully relaxed. You want

a (diverging) corrective lens which puts a virtual image of the source object there. So we can solve

the thin lens equation to find <math>f</math> for an arbitrary source object distance <math>s</math>

given <math>s' = -cm</math>.

For full correction, that's easy : <math>s=\infty</math> and so <math>f=s'</math>. E.g. if your

blur horizon is 20cm you need a -5D correction.

For differentials to use a screen at, say, 50cm, just use <math>\frac{1}{f} = \frac{1}{0.50} + -\frac{1}{0.20} = -3D</math>.

(Which is consistent with the previous version, adding 2D to the calculated full correction of -5D.)

==Point of refraction==

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==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]]

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

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>

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

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

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

~~<math>~~\frac{~~font\ height (mm)~~}{~~distance~~\ ~~to\ sign(m)~~} \times 13.~~75 = denominator~~ \times of \frac{20}{x}</math>

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

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

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

* 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)

~~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>

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.

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

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

= \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>

Then, for example

* 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>

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

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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 5: / Line 5: @@
 __TOC__
 {{clear}}
-==The thin lens equation==
+==Lenses==
+[[File:Lens3.svg|right]]
+The image shows a typical converging lens. The three rays drawn each have an interesting characteristic:
+* the top ray enters the lens parallel with the optic axis, and so passes through the focal point on the other side
+* the middle ray passes through the optical centre of the lens, and is undeviated
+* the bottom ray passes through the focal point on the incident side, and so emerges parallel to the optic axis
+===The thin lens equation===
 <math>\frac{1}{f} = (\frac{1}{s}) + (\frac{1}{s'})</math>
 where
 * <math>f</math> = focal length of lens
-* <math>s</math> = distance to object
+* <math>s</math> = distance to object (<math>S_1</math> on the picture)
-* <math>s'</math> = distance to image
+* <math>s'</math> = distance to image (<math>S_2</math> on the picture)
 ===Infinity===
 The term '''object at infinity''' is often used. When <math>s=\infty</math> is substituted into the thin lens equation, that term vanishes, so that the focal length is then just the image location. For any sufficiently large object distance, the contribution from the reciprocal becomes negligible.
+As the object approaches the lens, the outgoing rays converge less and less, until the object reaches the focus of the lens (<math>s=f</math>), and the transmitted light is parallel - we say it "focuses at <math>s'=\infty</math>", which means it never comes together into a focus. (This can also be described as a virtual image at <math>s'=-\infty</math> - see below.)
 ===Virtual image===
-A converging lens (such as a magnifying glass) behaves like a "typical" lens - the incoming light is brought to a focus on the opposite side of the lens.
-A diverging lens behaves differently - the light rays spreading from the source object are refracted outwards so that they are diverging even faster. They are not brought to a focus in any intuitive sense. Instead, the light behaves ''as if'' it was coming from a closer object. This is termed a '''virtual''' image - it lies between the source object and 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.
+When the object is brought even closer to the lens (<math>s < f</math>) the emerging rays are now diverging. When substituted into the equation, <math>s' < 0</math>. This is interpreted as a '''virtual image''', ''behind'' the lens. This is the mode in which reading glasses (plus lenses) are used - the eye is able to focus on the virtual image which appears to be further away than the real source object.
+===Diverging lens===
+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. (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.)
-The corrective lens for [[myopia]] is a diverging lens. It works by forming a virtual image of objects far away, and it is that virtual image that the near-sighted eye is able to focus on.
 ===Diopters are inverse meters===
@@ Line 33: / 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 57: / Line 69: @@
 But rather than actually placing such a lens in front of your full prescription,
-simply subtract that value.
+simply add that value (being careful with signs).
 ie if your screen is 50cm away, that corresponds to a power of <math>\frac{1}{0.5} = 2D</math>.
-So that's the value you'd subtract from your full prescription.
+So that's the value you'd add to your full prescription (resulting in a less-negative lens).
 ===From blur horizon of naked eye===
-From your cm measurement...
+Note that this does not take either [[cylinder]] ([[astigmatism]]) or [[vertex distance]] into account.
+Your [[cm measurement]] gives you the distance the eye can see when it is fully relaxed. You want
+a (diverging) corrective lens which puts a virtual image of the source object there. So we can solve
+the thin lens equation to find <math>f</math> for an arbitrary source object distance <math>s</math>
+given <math>s' = -cm</math>.
+For full correction, that's easy : <math>s=\infty</math> and so <math>f=s'</math>. E.g. if your
+blur horizon is 20cm you need a -5D correction.
+For differentials to use a screen at, say, 50cm, just use <math>\frac{1}{f} = \frac{1}{0.50} + -\frac{1}{0.20} = -3D</math>.
+(Which is consistent with the previous version, adding 2D to the calculated full correction of -5D.)
 ==Point of refraction==
@@ Line 71: / 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]]
-Note: 5Arcminutes = 20/20
+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>
-Set up proportion: <math>\frac{a}{(\frac{20}{x})} = \frac{5}{(\frac{20}{20})}</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:
-===Visual acuity (mm/metres)===
+<math> \text{Acuity} = 1/A = \frac{pi}{60 \times 180} \times \frac{b}{a} = 0.00029 \times \frac{b}{a}</math>
-<math>\frac{font\ height (mm)}{distance\ to\ sign(m)} \times 13.75 = denominator \times of \frac{20}{x}</math>
-===Visual acuity (in/feet)===
+An alternative representation of acuity is "logMAR" ("log<sub>10</sub> Minimum Angle of Resolution"). This has a bigger-is-worse direction.
-<math>\frac{font\ height(in)}{distance\ to\ sign(ft.)} \times 1146 = denominator \times of \frac{20}{x}</math>
+* 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)
-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>
+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.
+Acuity is reported in the form "distance/letter-row", such as "20/40", etc.
+<math> \text{Acuity} = 5 \times 0.00029 \times \frac{ \text{distance to chart} }{ \text{letter height} }
+= \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>
+Then, for example
+* 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>
+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 96: / 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}}