Endmyopia Wiki - User contributions [en]

Astigmatism

2020-09-17T09:00:11Z

Divenal: Mention Stenopaeic slit and its chromatic analogue.

'''Astigmatism''' is an eye condition that means you have blur in a specific direction, or [[axis]] (technically, depending on the notation used for your prescription, the axis may indicate the angle of the eye's meridian where you have the least focusing power, or the one where you have the most). Astigmatism is compensated with [[cylinder]] lenses. A cylinder lens adds power along one particular meridian of the eye.

Astigmatism often reduces spontaneously as myopia is corrected.

==Understanding astigmatism==

Astigmatism is caused by an irregularly shaped cornea or lens. The first is called corneal astigmatism, which is the more common form, and the second is called lenticular astigmatism. "Regular" astigmatism is often described as a having the cornea shaped like a rugby ball, rather than spherical like a basketball. The optics of an ''idealised'' lens of such a form would cause incoming light on different planes (corresponding to the two principal axes of the lens) to be focused at different offsets beyond the lens.

[[File:Astigmatism.svg|Astigmatism]]

Most diagrams of lenses show only a single vertical plane. In the real world, of course, there is a full cone of light arriving on the lens from the source object. This diagram shows two different cross-sections through the incident cone of light, aligned with the two axes. Rays in the horizontal cross-section (which contains the stronger curve) are focused earlier - at the label 'T' (for 'Tangential'). The rays in the vertical cross-section (the 'Saggital') are focused further behind, at 'S'. The other rays around the light cone are focused at points in between the two, giving an image smeared out along the axis.

===Stenopaeic slit===

The Stenopaeic slit is a simple tool which can be used in diagnosis / measurement of astigmatism. It is simply a disk with a narrow slit which can be rotated to find the clearest image. By reducing light coming in the "wrong" direction, the spherical correction on each axis can be measured directly.

===Analogy with Chromatic Aberration===

It may be simpler to picture the effect by comparing with [[Chromatic Aberration]]. In both cases, an extra variable means different parts of the light are focused differently.

[[File:Chromatic aberration lens diagram.svg|Chromatic aberration lens diagram]]

(Unfortunately the astigmatism diagram choose red and blue the wrong way round !)

* If the red light is focused on the retina, the green/blue light is focused in front, and is blurred.
* If the blue light is focused correctly, green and red are focused beyond the retina and is blurred.
* It's not possible to get everything into focus using only spherical lenses.
As a compromise, focusing the green light on the retina causes a little bit of myopic blue blur and hyperopic red blur. This corresponds to the "circle of least confusion" in astigmatism.

On a prescription, there are two different conventions for specifying the cylinder. This corresponds to either quoting the spherical correction to focus red on the retina, and then the additional "minus" required to focus blue; or a spherical correction for blue, and then how much that can be reduced by for red. The average of the two, or the "spherical equivalence", is then the correction required to put green on the retina.

The analogue of the Stenopaeic slit is this model is a simple coloured filter : by allowing only monochromatic light into the eye, the spherical correction for each colour can be measured separately.

If the object is moved away, beyond your blur horizon, so that your eye can no longer keep the green light focused,
all colours will suffer myopic blur, but blue will have the worst blur. This corresponds to the directional blur
in astigmatism. Adding some spherical correction would allow you to push the green back into focus.

Chromatic aberration could be treated by adding some material which applies the opposite chromatic error - bending the blue light out a bit more than the red light, to cancel the error introduced by the eye. This corresponds to cylinder correction. (But as with all analogies, it's starting to stretch a bit thin...)

==Irregular Astigmatism==

A real cornea, of course, doesn't conform to expectations. Being messier, it just has a bulge, which means that the image is not only smeared out along the axis, but is rotated, resulting in multiple (blurred) images being perceived on the retina.

[[File:Astigmatism (Eye).png|Astigmatism (Eye)]]

==Childhood Astigmatism==
Astigmatism in young children often changes after they reach school age.<ref>{{Cite journal |last=Dobson |first=V. |last2=Fulton |first2=A. B. |last3=Sebris |first3=S. L. |date=1984-01-01 |title=Cycloplegic refractions of infants and young children: the axis of astigmatism. |url=https://iovs.arvojournals.org/article.aspx?articleid=2159731 |journal=Investigative Ophthalmology & Visual Science |language=en |volume=25 |issue=1 |pages=83–87 |issn=1552-5783}}</ref>

==Reducing astigmatism==
As with myopia, astigmatism should be tackled in small steps when selecting lenses for [[differential]] or [[normalized]] glasses. If only a small amount of cylinder correction is present, say 0.25 [[diopters]], the cylinder correction can be dropped, with no other changes. Otherwise, cylinder should be reduced in small increments. If sphere is being reduced, cylinder should not be changed at the same time, and conversely, if cylinder is being reduced then sphere should not be changed. The only time that both sphere and cylinder should be changed, is when converting to the spherical equivalent.

The link between astigmatism strength and visual acuity is weak. How your eyes and [[visual cortex]] respond to astigmatism is a greater factor in visual acuity than the number of dipoters.<ref>{{Cite journal |last=Remón |first=Laura |last2=Tornel |first2=Marta |last3=Furlan |first3=Walter D. |date=2006-05 |title=Visual Acuity in Simple Myopic Astigmatism: Influence of Cylinder Axis |url=https://journals.lww.com/optvissci/Abstract/2006/05000/Visual_Acuity_in_Simple_Myopic_Astigmatism_.11.aspx |journal=Optometry and Vision Science |language=en-US |volume=83 |issue=5 |pages=311–315 |doi=10.1097/01.opx.0000216099.29968.36 |issn=1538-9235}}</ref> If the standard advice for correcting astigmatism does not work for you, you may need a more gradual reduction.

The '''spherical equivalent''' of cylinder lenses may be useful, to simplify the reduced lens path. 0.50 cylinder means the power varies from 0D on one axis to 0.5D on the perpendicular axis. This can be substituted by the spherical power with the averaged value of 0.25D. For example, a full prescription of "-1.00 Sphere -1.50 Cylinder" could be converted to "-1.75 Sphere". The resulting spherical equivalent is not intended to compensate for the asymmetry of the lens, so it will introduce some directional blur.

Regardless of how the myope reduces their lenses, the goal of each reduction is to have a small amount of "useful blur", to be cleared up with [[active focus]] and good habits.

==Resources==
* [https://endmyopia.org/the-definitive-guide-astigmatism/ EndMyopia Blog - The Definitive Guide: What Is Astigmatism]
* [https://endmyopia.org/tag/astigmatism-2/ EndMyopia Blog - all astigmatism articles]
* [https://endmyopia.org/diy-tools-how-to-measure-your-astigmatism-diopters/ How To Measure Your Astigmatism Diopters]
* [https://visiontools.netlify.app/ Vision tool]
* [https://community.endmyopia.org/t/having-trouble-figuring-out-my-astigmatism/4843/6 Astigmatism Assasin's guide]

==References==
{{reflist}}

[[Category:Articles]]
[[Category:Eye conditions]]

Astigmatism

2020-08-29T09:31:37Z

Divenal: /* Analogy with Chromatic Aberration */ add analogy for the two different ways to prescribe cylinder.

'''Astigmatism''' is a really annoying eye condition that means you have blur in a specific direction, or [[axis]] (technically, depending on the notation used for your prescription, the axis may indicate the angle of the eye's meridian where you have the least focusing power, or the one where you have the most). Astigmatism is compensated with [[cylinder]] lenses. A cylinder lens adds power along one particular meridian of the eye.

Astigmatism often reduces spontaneously as myopia is corrected.

==Understanding astigmatism==

Astigmatism is caused by an irregularly shaped cornea or lens. The first is called corneal astigmatism, which is the more common form, and the second is called lenticular astigmatism. "Regular" astigmatism is often described as a having the cornea shaped like a rugby ball, rather than spherical like a basketball. The optics of an ''idealised'' lens of such a form would cause incoming light on different planes (corresponding to the two principal axes of the lens) to be focused at different offsets beyond the lens.

[[File:Astigmatism.svg|Astigmatism]]

Most diagrams of lenses show only a single vertical plane. In the real world, of course, there is a full cone of light arriving on the lens from the source object. This diagram shows two different cross-sections through the incident cone of light, aligned with the two axes. Rays in the horizontal cross-section (which contains the stronger curve) are focused earlier - at the label 'T' (for 'Tangential'). The rays in the vertical cross-section (the 'Saggital') are focused further behind, at 'S'. The other rays around the light cone are focused at points in between the two, giving an image smeared out along the axis.

===Analogy with Chromatic Aberration===

It may be simpler to picture the effect by comparing with [[Chromatic Aberration]]. In both cases, an extra variable means different parts of the light are focused differently.

[[File:Chromatic aberration lens diagram.svg|Chromatic aberration lens diagram]]

(Unfortunately the astigmatism diagram choose red and blue the wrong way round !)

* If the red light is focused on the retina, the green/blue light is focused in front, and is blurred.
* If the blue light is focused correctly, green and red are focused beyond the retina and is blurred.
* It's not possible to get everything into focus using only spherical lenses.
As a compromise, focusing the green light on the retina causes a little bit of myopic blue blur and hyperopic red blur. This corresponds to the "circle of least confusion" in astigmatism.

On a prescription, there are two different conventions for specifying the cylinder. This corresponds to either quoting the spherical correction to focus red on the retina, and then the additional "minus" required to focus blue; or a spherical correction for blue, and then how much that can be reduced by for red. The average of the two, or the "spherical equivalence", is then the correction required to put green on the retina.

If the object is moved away, beyond your blur horizon, so that your eye can no longer keep the green light focused,
all colours will suffer myopic blur, but blue will have the worst blur. This corresponds to the directional blur
in astigmatism. Adding some spherical correction would allow you to push the green back into focus.

Chromatic aberration could be treated by adding some material which applies the opposite chromatic error - bending the blue light out a bit more than the red light, to cancel the error introduced by the eye. This corresponds to cylinder correction. (But as with all analogies, it's starting to stretch a bit thin...)

==Irregular Astigmatism==

A real cornea, of course, doesn't conform to expectations. Being messier, it just has a bulge, which means that the image is not only smeared out along the axis, but is rotated, resulting in multiple (blurred) images being perceived on the retina.

[[File:Astigmatism (Eye).png|Astigmatism (Eye)]]

==Childhood Astigmatism==
Astigmatism in young children often changes after they reach school age.<ref>{{Cite journal |last=Dobson |first=V. |last2=Fulton |first2=A. B. |last3=Sebris |first3=S. L. |date=1984-01-01 |title=Cycloplegic refractions of infants and young children: the axis of astigmatism. |url=https://iovs.arvojournals.org/article.aspx?articleid=2159731 |journal=Investigative Ophthalmology & Visual Science |language=en |volume=25 |issue=1 |pages=83–87 |issn=1552-5783}}</ref>

==Reducing astigmatism==
As with myopia, astigmatism should be tackled in small steps when selecting lenses for [[differential]] or [[normalized]] glasses. If only a small amount of cylinder correction is present, say 0.25 [[diopters]], the cylinder correction can be dropped, with no other changes. Otherwise, cylinder should be reduced in small increments. If sphere is being reduced, cylinder should not be changed at the same time, and conversely, if cylinder is being reduced then sphere should not be changed. The only time that both sphere and cylinder should be changed, is when converting to the spherical equivalent.

The link between astigmatism strength and visual acuity is weak. How your eyes and [[visual cortex]] respond to astigmatism is a greater factor in visual acuity than the number of dipoters.<ref>{{Cite journal |last=Remón |first=Laura |last2=Tornel |first2=Marta |last3=Furlan |first3=Walter D. |date=2006-05 |title=Visual Acuity in Simple Myopic Astigmatism: Influence of Cylinder Axis |url=https://journals.lww.com/optvissci/Abstract/2006/05000/Visual_Acuity_in_Simple_Myopic_Astigmatism_.11.aspx |journal=Optometry and Vision Science |language=en-US |volume=83 |issue=5 |pages=311–315 |doi=10.1097/01.opx.0000216099.29968.36 |issn=1538-9235}}</ref> If the standard advice for correcting astigmatism does not work for you, you may need a more gradual reduction.

The '''spherical equivalent''' of cylinder lenses may be useful, to simplify the reduced lens path. 0.50 cylinder means the power varies from 0D on one axis to 0.5D on the perpendicular axis. This can be substituted by the spherical power with the averaged value of 0.25D. For example, a full prescription of "-1.00 Sphere -1.50 Cylinder" could be converted to "-1.75 Sphere". The resulting spherical equivalent is not intended to compensate for the asymmetry of the lens, so it will introduce some directional blur.

Regardless of how the myope reduces their lenses, the goal of each reduction is to have a small amount of "useful blur", to be cleared up with [[active focus]] and good habits.

==Resources==
*[https://endmyopia.org/the-definitive-guide-astigmatism/ EndMyopia Blog - The Definitive Guide: What Is Astigmatism]
*[https://endmyopia.org/tag/astigmatism-2/ EndMyopia Blog - all astigmatism articles]

==References==
{{reflist}}

[[Category:Articles]]
[[Category:Eye conditions]]

Astigmatism

2020-08-28T15:00:21Z

Divenal: Expand a bit on the analogy with chromatic aberration, which I find a lot easier to fit into my head.

'''Astigmatism''' is a really annoying eye condition that means you have blur in a specific direction, or [[axis]] (technically, depending on the notation used for your prescription, the axis may indicate the angle of the eye's meridian where you have the least focusing power, or the one where you have the most). Astigmatism is compensated with [[cylinder]] lenses. A cylinder lens adds power along one particular meridian of the eye.

Astigmatism often reduces spontaneously as myopia is corrected.

==Understanding astigmatism==

Astigmatism is caused by an irregularly shaped cornea or lens. The first is called corneal astigmatism, which is the more common form, and the second is called lenticular astigmatism. "Regular" astigmatism is often described as a having the cornea shaped like a rugby ball, rather than spherical like a basketball. The optics of an ''idealised'' lens of such a form would cause incoming light on different planes (corresponding to the two principal axes of the lens) to be focused at different offsets beyond the lens.

[[File:Astigmatism.svg|Astigmatism]]

Most diagrams of lenses show only a single vertical plane. In the real world, of course, there is a full cone of light arriving on the lens from the source object. This diagram shows two different cross-sections through the incident cone of light, aligned with the two axes. Rays in the horizontal cross-section (which contains the stronger curve) are focused earlier - at the label 'T' (for 'Tangential'). The rays in the vertical cross-section (the 'Saggital') are focused further behind, at 'S'. The other rays around the light cone are focused at points in between the two, giving an image smeared out along the axis.

===Analogy with Chromatic Aberration===

It may be simpler to picture the effect by comparing with [[Chromatic Aberration]]. In both cases, an extra variable means different parts of the light are focused differently.

[[File:Chromatic aberration lens diagram.svg|Chromatic aberration lens diagram]]

(Unfortunately the astigmatism diagram choose red and blue the wrong way round !)

* If the red light is focused on the retina, the green/blue light is focused in front, and is blurred.
* If the blue light is focused correctly, green and red are focused beyond the retina and is blurred.
* It's not possible to get everything into focus using only spherical lenses.
As a compromise, focusing the green light on the retina causes a little bit of myopic blue blur and hyperopic red blur. This corresponds to the "circle of least confusion" in astigmatism.

If the object is moved away, beyond your blur horizon, so that your eye can no longer keep the green light focused,
all colours will suffer myopic blur, but blue will have the worst blur. This corresponds to the directional blur
in astigmatism. Adding some spherical correction would allow you to push the green back into focus.
* The required "spherical equivalence" is the average of corrections required to bring blue or red into focus.

Chromatic aberration could be treated by adding some material which applies the opposite chromatic error - bending the blue light out a bit more than the red light, to cancel the error introduced by the eye. This corresponds to cylinder correction. (But as with all analogies, it's starting to stretch a bit thin...)

==Irregular Astigmatism==

A real cornea, of course, doesn't conform to expectations. Being messier, it just has a bulge, which means that the image is not only smeared out along the axis, but is rotated, resulting in multiple (blurred) images being perceived on the retina.

[[File:Astigmatism (Eye).png|Astigmatism (Eye)]]

==Childhood Astigmatism==
Astigmatism in young children often changes after they reach school age.<ref>{{Cite journal |last=Dobson |first=V. |last2=Fulton |first2=A. B. |last3=Sebris |first3=S. L. |date=1984-01-01 |title=Cycloplegic refractions of infants and young children: the axis of astigmatism. |url=https://iovs.arvojournals.org/article.aspx?articleid=2159731 |journal=Investigative Ophthalmology & Visual Science |language=en |volume=25 |issue=1 |pages=83–87 |issn=1552-5783}}</ref>

==Reducing astigmatism==
As with myopia, astigmatism should be tackled in small steps when selecting lenses for [[differential]] or [[normalized]] glasses. If only a small amount of cylinder correction is present, say 0.25 [[diopters]], the cylinder correction can be dropped, with no other changes. Otherwise, cylinder should be reduced in small increments. If sphere is being reduced, cylinder should not be changed at the same time, and conversely, if cylinder is being reduced then sphere should not be changed. The only time that both sphere and cylinder should be changed, is when converting to the spherical equivalent.

The link between astigmatism strength and visual acuity is weak. How your eyes and [[visual cortex]] respond to astigmatism is a greater factor in visual acuity than the number of dipoters.<ref>{{Cite journal |last=Remón |first=Laura |last2=Tornel |first2=Marta |last3=Furlan |first3=Walter D. |date=2006-05 |title=Visual Acuity in Simple Myopic Astigmatism: Influence of Cylinder Axis |url=https://journals.lww.com/optvissci/Abstract/2006/05000/Visual_Acuity_in_Simple_Myopic_Astigmatism_.11.aspx |journal=Optometry and Vision Science |language=en-US |volume=83 |issue=5 |pages=311–315 |doi=10.1097/01.opx.0000216099.29968.36 |issn=1538-9235}}</ref> If the standard advice for correcting astigmatism does not work for you, you may need a more gradual reduction.

The '''spherical equivalent''' of cylinder lenses may be useful, to simplify the reduced lens path. 0.50 cylinder means the power varies from 0D on one axis to 0.5D on the perpendicular axis. This can be substituted by the spherical power with the averaged value of 0.25D. For example, a full prescription of "-1.00 Sphere -1.50 Cylinder" could be converted to "-1.75 Sphere". The resulting spherical equivalent is not intended to compensate for the asymmetry of the lens, so it will introduce some directional blur.

Regardless of how the myope reduces their lenses, the goal of each reduction is to have a small amount of "useful blur", to be cleared up with [[active focus]] and good habits.

==Resources==
*[https://endmyopia.org/the-definitive-guide-astigmatism/ EndMyopia Blog - The Definitive Guide: What Is Astigmatism]
*[https://endmyopia.org/tag/astigmatism-2/ EndMyopia Blog - all astigmatism articles]

==References==
{{reflist}}

[[Category:Articles]]
[[Category:Eye conditions]]

Talk:Optics related math

2020-08-07T17:24:56Z

Divenal:

{{re|NottNott}}: the server frequently gives error "Math extension cannot connect to Restbase" when attempting to render the math mode stuff. Seems like it's a common issue, eg
* https://www.mediawiki.org/wiki/Topic:Tqkr5mdviskc8fmk
* https://www.mediawiki.org/wiki/Topic:Uo3kkmmop1jj9vhw
[[User:Divenal|Divenal]] ([[User talk:Divenal|talk]]) 15:54, 5 August 2020 (UTC)
:{{re|Divenal}} How do you reproduce the bug? It may or may not be fixable from what I'm reading :) -[[User:NottNott|<span style="color:#e67e22">NottNott</span>]] <small>([[User talk:NottNott|talk]])</small> 17:46, 5 August 2020 (UTC)
:{{re|NottNott}}: Trying just now... if I edit this article (optics related math) and press preview (without making any changes) I get the error. Hmm - but then trying again, it was fine. [[User:Divenal|Divenal]] ([[User talk:Divenal|talk]])

Talk:Optics related math

2020-08-07T17:21:36Z

Divenal:

{{re|NottNott}}: the server frequently gives error "Math extension cannot connect to Restbase" when attempting to render the math mode stuff. Seems like it's a common issue, eg
* https://www.mediawiki.org/wiki/Topic:Tqkr5mdviskc8fmk
* https://www.mediawiki.org/wiki/Topic:Uo3kkmmop1jj9vhw
[[User:Divenal|Divenal]] ([[User talk:Divenal|talk]]) 15:54, 5 August 2020 (UTC)
:{{re|Divenal}} How do you reproduce the bug? It may or may not be fixable from what I'm reading :) -[[User:NottNott|<span style="color:#e67e22">NottNott</span>]] <small>([[User talk:NottNott|talk]])</small> 17:46, 5 August 2020 (UTC)
:{{re|NottNott}}: Trying just now... if I edit this article (optics related math) and press preview (without making any changes) I get the error.[[User:Divenal|Divenal]] ([[User talk:Divenal|talk]])

User:Divenal/sandbox

2020-08-05T16:04:31Z

Divenal:

Welcome, {{#ifanon:anonymous|registered}} user.

[[Wikipedia:Scalable_Vector_Graphics|SVG]] is a simple xml markup system for describing images. (Well, simple for the trivial stuff I'm using it for...) You just tell it "I want a blue circle here, a red square there, ...". Unlike a bitmap, this will render perfectly at any scale. But obviously you need software to turn it into bitmap of the size you need.

The wiki will do so itself. You can add a reference to the image from your own wiki page and tell it what size to display. Just add a reference of the form <nowiki>[[File:filename.svg|512px]]</nowiki>. The 512px is the image size you want, and you can set that to whatever you want. (I'm not quite sure if it renders it for each size, or if it renders it once on upload and just rescales the bitmap. Not sure if the wiki has an option to just pass the svg directly to the browser for rendering..?) see https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Images

Modern browsers can generate svg directly. If you click on one of the images, it will take you to the wiki page about the file. Clicking again should present the svg directly to the browser for local rendering. Resizing there should be accurate since it should re-render from the vector description. You can use the browser's view-source option to see the (trivial) svg that creates the image. It should be fairly obvious how it works. The only slightly non-trivial thing is the <nowiki><g ...></nowiki> stuff. I tend to use that to move the origin to the centre of the canvas, and various other convenient defaults, to save having to repeat stuff on every line.

There are some online convertors you can use. Just google.

If you want to do it locally, one free application (both windows and linux) is [[wikipedia:Inkscape|Inkscape]]. You can either load the svg file interactively and then export via the menus, or use a command line:
inkscape -w 1024 -h 1024 input.svg --export-filename output.png

Another free command-line tool is [[wikipedia::ImageMagick|ImageMagick]]
convert -size 1024x1024 input.svg output.png

----

[[File:Dd_blur_tool.svg|480px|thumb|right|click to enlarge]]

This is a tool that might be useful for distingushing https://endmyopia.org/astigmatic-cylinder-blur-vs-myopic-spherical-blur/

The gaps are intended to be calibrated to 1 minute of arc, equivalent to the gaps in landolt C test. The idea was that directional blur will close some of the gaps, but "blurry blur" will do them all. So when measuring cm's, you can distinguish the two.

Also the distance at which ghosting first appears, if you're also measuring that. For me, the ghosting appears in particular directions, so you can concentrate on the gaps on the lines where the ghost image is most-separated from the original. (ie if ghosting happens in a north-east direction, look at the gaps on the lines going north-west.)

<br clear="both" />
----

[[File:rgfocus.svg|256px|thumb|right]]

This is a quick svg hack-up of the test page on [[Varakari's_Vision_Log_Tool]]. My intention was to be able to display it on my smartphone for measurement, but should work on any browser.

Do feel free to improve it.

<br clear="both" />
----

[[File:rgsunrise.svg|384px|thumb|right]]

This combines the two ideas... It's sized to suit my smartphone, but feel free to make your custom version as above.

<br clear="both" />

----

[[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)

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>

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.

Visual acuity

2020-08-05T15:55:27Z

Divenal: /* See Also */ add ref to the maths page

Visual acuity is the measurement of how well the vision system as a whole is recognizing what it sees. This is typically measured with a [[Snellen chart]] and expressed as the distance you need to be from the chart to recognize the characters over the distance that someone with normal vision can recognize the characters.

==Reading a Snellen chart==
{{main|Snellen chart}}
* 10/400 would be very bad (possibly just [[myopic]]),
* 20/20 is (by definition) normal, and
* 40/20 is excellent.

Visual acuity for [[myopes]] is typically measured at [[distance vision|20 feet or 6 meters]]. Lens or mirror tricks may be used to change the effective test distance in a smaller room. Visual acuity for [[hyperopes]] is typically measured at 36cm, and a [[presbyope]] would be measured at both distances.

With text that we are familiar with, the brain may clear up that text more than our vision would actually allow.

==Reasons differing visual acuity==
[[Refraction]] is only one part of visual acuity. Various medical conditions can cause physical blockage/dispersion of light in the eye, problems detecting light in the eye, or problems with the visual processing that turns a series of electrochemical signals into a picture in our mind's eye. See an [[optometrist]] if your vision can't be corrected with [[refraction]].

=See Also=
* [[Emmetropia]]
* [[Optics_related_math#Visual_acuity]]

==References==
{{reflist}}

[[Category:Articles]]

Talk:Optics related math

2020-08-05T15:54:06Z

Divenal: Created page with "{{re|NottNott}}: the server frequently gives error "Math extension cannot connect to Restbase" when attempting to render the math mode stuff. Seems like it's a common issue, e..."

Optics related math

2020-08-05T15:26:00Z

Divenal: /* Diopters are inverse meters */ Seems that one should use \text{...} inside math mode to render plain text.

[[File:Math guy with glasses.gif|right]]
Here's a page with maths related to diopters and glasses.

You don't really need to know any of this stuff to improve your eyesight, but it's good to know for deeper understanding {{smiley}}
__TOC__
{{clear}}
==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_1</math> on the picture)
* <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===

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.)

===Diopters are inverse meters===

Intuitively, the more powerful a lens is, the more rapidly it can bring incoming light to a focus. So the power is defined as the inverse of the focal length.

''See Also [[Diopters]]''

''See Also [[cm Measurement]]''

Remember that 100cm = 1m.

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

conversely

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

==Calculating correction==

How to calculate the strength of corrective lenses

===From 20/20 prescription===

The advantage of calculating the strength of [[differentials]]
as a reduction from full prescription is that you don't have to worry about
the [[cylinder]] element - just preserve that part.

Assuming you are wearing your full 20/20 correction (or even if you are fortunate
to be [[emmetropic]]), the eye wants to receive parallel incident light to be relaxed.
So we can use a simple geometric argument to calculate the strength of the (converging)
lens which should be interposed : an object placed at the focus of a converging
lens will result in parallel outgoing light ("focus at infinity"). This parallel light
is exactly what the (fully corrected) eye wants to receive.

But rather than actually placing such a lens in front of your full prescription,
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 add to your full prescription (resulting in a less-negative lens).

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

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==
''See also [[Refraction]]''
{{#ev:youtube|9z2jjT_Gm7o|400}}

==Visual acuity==
[[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)

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==
* The typical emmetropic eye is 25mm
* change in axial length of 1mm = 3D

If someone with typical eyes wanted to adapt say 20/20 to .25 less
normalized within 3-4 months
would need to decrease axial length 0.083mm
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)]]

==References==
{{reflist}}

Optics related math

2020-08-05T15:24:44Z

Divenal: Updating the visual acuity section. Unfortunately, the wiki frequently gets errors when contacting the server that renders the maths stuff.

[[File:Math guy with glasses.gif|right]]
Here's a page with maths related to diopters and glasses.

You don't really need to know any of this stuff to improve your eyesight, but it's good to know for deeper understanding {{smiley}}
__TOC__
{{clear}}
==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_1</math> on the picture)
* <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===

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.)

===Diopters are inverse meters===

Intuitively, the more powerful a lens is, the more rapidly it can bring incoming light to a focus. So the power is defined as the inverse of the focal length.

''See Also [[Diopters]]''

''See Also [[cm Measurement]]''

Remember that 100cm = 1m.

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

conversely

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

==Calculating correction==

How to calculate the strength of corrective lenses

===From 20/20 prescription===

The advantage of calculating the strength of [[differentials]]
as a reduction from full prescription is that you don't have to worry about
the [[cylinder]] element - just preserve that part.

Assuming you are wearing your full 20/20 correction (or even if you are fortunate
to be [[emmetropic]]), the eye wants to receive parallel incident light to be relaxed.
So we can use a simple geometric argument to calculate the strength of the (converging)
lens which should be interposed : an object placed at the focus of a converging
lens will result in parallel outgoing light ("focus at infinity"). This parallel light
is exactly what the (fully corrected) eye wants to receive.

But rather than actually placing such a lens in front of your full prescription,
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 add to your full prescription (resulting in a less-negative lens).

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

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==
''See also [[Refraction]]''
{{#ev:youtube|9z2jjT_Gm7o|400}}

==Visual acuity==
[[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)

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==
* The typical emmetropic eye is 25mm
* change in axial length of 1mm = 3D

If someone with typical eyes wanted to adapt say 20/20 to .25 less
normalized within 3-4 months
would need to decrease axial length 0.083mm
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)]]

==References==
{{reflist}}

2020-07-22T09:38:50Z

Divenal:

2020-07-15T16:48:40Z

Divenal: Divenal uploaded a new version of File:Rgfocus.svg

== Summary ==
Quick svg hackup of varakari's focus page
== Licensing ==
{{self|cc-by-sa-4.0}}

User:Divenal/sandbox

2020-07-15T12:10:32Z

Divenal: Updates to reflected new image size of 256x256

File:Rgfocus.svg

2020-07-15T12:07:25Z

Divenal: Divenal uploaded a new version of File:Rgfocus.svg

== Summary ==
Quick svg hackup of varakari's focus page
== Licensing ==
{{self|cc-by-sa-4.0}}

User:Divenal/sandbox

2020-07-15T10:41:08Z

Divenal:

Welcome, {{#ifanon:anonymous|registered}} user.

[[File:Dd_blur_tool.svg|200px|thumb|right|click to enlarge]]

This is a tool that might be useful for distingushing https://endmyopia.org/astigmatic-cylinder-blur-vs-myopic-spherical-blur/

The gaps are intended to be calibrated to 1 minute of arc, equivalent to the gaps in landolt C test. The idea was that directional blur will close some of the gaps, but "blurry blur" will do them all. So when measuring cm's, you can distinguish the two.

Also the distance at which ghosting first appears, if you're also measuring that. For me, the ghosting appears in particular directions, so you can concentrate on the gaps on the lines where the ghost image is most-separated from the original. (ie if ghosting happens in a north-east direction, look at the gaps on the lines going north-west.)

----

[[File:rgfocus.svg|250px|thumb|right]]

This is a quick svg hack-up of the test page on [[Varakari's_Vision_Log_Tool]]. My intention was to be able to display it on my smartphone for measurement, but should work on any browser.

The image itself is svg, and so infinitely scalable. But the wiki tends to generate it as a png. So if you need to resize it, it's better to do it as svg rather than zooming in or out of the png. One simple way to do that is to add a reference to the image from your own wiki page, and adjust the size. Then the wiki will generate a png at that size, and you can download it locally.

To do so, just add a reference of the form <nowiki>[[File:rgfocus.svg|250px]]</nowiki>. The 250px is the image size, and you can set that to whatever you want. There are various other options that can go in - see https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Images

If you click on the image, it will take to the page about the file. Clicking again should present the raw svg to the browser. You can use the browser's view-source option to see the (trivial) svg that creates the image. It should be fairly obvious how it works. The only slightly non-trivial thing is the <nowiki><g ...></nowiki> stuff. That just moves the origin to the centre of the canvas and sets the default text anchor, so that all the entities have an x-co-ord of 0. Just saves having to repeat stuff on every line.

Do feel free to improve it.

File:Rgfocus.svg

2020-07-14T21:16:05Z

Divenal: Divenal uploaded a new version of File:Rgfocus.svg

== Summary ==
Quick svg hackup of varakari's focus page
== Licensing ==
{{self|cc-by-sa-4.0}}

User:Divenal/sandbox

2020-07-14T19:57:40Z

Divenal:

2020-06-29T08:46:27Z

Divenal:

Optics related math

2020-06-27T18:52:46Z

Divenal: /* Infinity */ minor rewording, to paint a picture of the object moving towards the lens continuously, rather than just jumping straight from infinity to the focus.

[[File:Math guy with glasses.gif|right]]
Here's a page with maths related to diopters and glasses.

You don't really need to know any of this stuff to improve your eyesight, but it's good to know for deeper understanding {{smiley}}
__TOC__
{{clear}}
==The thin lens equation==

[[File:Lens3.svg|right]]
<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_1</math> on the picture)
* <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.

===Virtual image===

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.

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.

===Diopters are inverse meters===

Intuitively, the more powerful a lens is, the more rapidly it can bring incoming light to a focus. So the power is defined as the inverse of the focal length.

''See Also [[Diopters]]''

''See Also [[cm Measurement]]''

Remember that 100cm = 1m.

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

conversely

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

==Calculating correction==

How to calculate the strength of corrective lenses

===From 20/20 prescription===

The advantage of calculating the strength of [[differentials]]
as a reduction from full prescription is that you don't have to worry about
the [[cylinder]] element - just preserve that part.

Assuming you are wearing your full 20/20 correction (or even if you are fortunate
to be [[emmetropic]]), the eye wants to receive parallel incident light to be relaxed.
So we can use a simple geometric argument to calculate the strength of the (converging)
lens which should be interposed : an object placed at the focus of a converging
lens will result in parallel outgoing light ("focus at infinity"). This parallel light
is exactly what the (fully corrected) eye wants to receive.

But rather than actually placing such a lens in front of your full prescription,
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 add to your full prescription (resulting in a less-negative lens).

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

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==
''See also [[Refraction]]''
{{#ev:youtube|9z2jjT_Gm7o|400}}

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

Note: 5Arcminutes = 20/20

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

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

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

==Average axial length accomodation/rate of change==
<math>typical\ emmetropic\ eye = 25mm = 25,000\ microns</math>

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

If someone with typical eyes wanted to adapt say 20/20 to .25 less
normalized within 3-4 months
would need to decrease axial length 0.083mm
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)]]

==References==
{{reflist}}

Optics related math

2020-06-27T16:34:43Z

Divenal: /* The thin lens equation */ mention converging lens with s < f. Could do with some more pictures, really...

[[File:Math guy with glasses.gif|right]]
Here's a page with maths related to diopters and glasses.

You don't really need to know any of this stuff to improve your eyesight, but it's good to know for deeper understanding {{smiley}}
__TOC__
{{clear}}
==The thin lens equation==

[[File:Lens3.svg|right]]
<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_1</math> on the picture)
* <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.

Similarly, if the object is at the focus of the lens (<math>s=f</math>), the transmitted light is parallel - we say it "focuses at <math>s'=\infty</math>", which means it never comes together into a focus.

===Virtual image===

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.

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.

===Diopters are inverse meters===

Intuitively, the more powerful a lens is, the more rapidly it can bring incoming light to a focus. So the power is defined as the inverse of the focal length.

''See Also [[Diopters]]''

''See Also [[cm Measurement]]''

Remember that 100cm = 1m.

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

conversely

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

==Calculating correction==

How to calculate the strength of corrective lenses

===From 20/20 prescription===

The advantage of calculating the strength of [[differentials]]
as a reduction from full prescription is that you don't have to worry about
the [[cylinder]] element - just preserve that part.

Assuming you are wearing your full 20/20 correction (or even if you are fortunate
to be [[emmetropic]]), the eye wants to receive parallel incident light to be relaxed.
So we can use a simple geometric argument to calculate the strength of the (converging)
lens which should be interposed : an object placed at the focus of a converging
lens will result in parallel outgoing light ("focus at infinity"). This parallel light
is exactly what the (fully corrected) eye wants to receive.

But rather than actually placing such a lens in front of your full prescription,
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 add to your full prescription (resulting in a less-negative lens).

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

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==
''See also [[Refraction]]''
{{#ev:youtube|9z2jjT_Gm7o|400}}

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

Note: 5Arcminutes = 20/20

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

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

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

==Average axial length accomodation/rate of change==
<math>typical\ emmetropic\ eye = 25mm = 25,000\ microns</math>

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

If someone with typical eyes wanted to adapt say 20/20 to .25 less
normalized within 3-4 months
would need to decrease axial length 0.083mm
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)]]

==References==
{{reflist}}

Eyeballs

2020-06-26T21:18:46Z

Divenal: /* Parts of the eye */ update rod/cone stuff

== Parts of the eye ==
[[File:Schematic diagram of the human eye en.svg|Schematic diagram of the human eye en|right]]
* '''Sclera''' - The white of the eye
* '''Cornea''' is the clear outer layer of the eye through which you can see the [[Iris]] and [[Pupil]]. It has an [[Index of Refraction]] of 1.376<ref name="Scale Model of Eye">{{Cite web |title=Scale Model of Eye |url=http://hyperphysics.phy-astr.gsu.edu/hbase/vision/eyescal.html |last=Nave |first=R |date=2020-05-25 |website=HyperPhysics}}</ref>, and a curved outer surface, contributing to the [[refractive state]] of the eye. The Cornea provides about 80% of the eye's total refracting power. If you have [[LASIK]] or [[PRK]] surgery it thins the cornea to change your refractive state.
* '''Aqueous humor''' - the fluid supporting the cornea
* '''Pupil''' - the hole where light enters the eye
* '''Iris''' - the Iris is the colored part of the eyeball that contains the muscles that control the opening size of the pupil.
* '''Ciliary muscle''' is a ring of muscle fibers in the eye that control the tendons supporting the natural [[lens]] of the eye, and controls the flow of [[aqueous humor]] behind the [[cornea]]. The Ciliary muscle is controlled by the Ciliary ganglion, which is a complex intersection of several nerve systems. The action of the ciliary muscle is the primary source of [[accommodation]] and [[ciliary spasm]] which causes [[pseudomyopia]].
* '''Lens''' - The part that changes the focus distance of the eye
* '''Rods and cones''' - Rods and Cones are the sensory cells in the back of your eye that detect light.
** Rods sense only light intensity, not color. They require lower levels of light to trigger, and so work better in low-light conditions. They are more sensitive to movement, and tend to be concentrated on the periphery of the retina. If you are outside at dusk, you may feel a sudden switch of your vision from color vision to black and white, this is your [[visual cortex]] switching to only rod input when cone input isn't working as well in dim light.
** Cones are the cells that detect color in your eye, but require much higher light levels to trigger. They concentrated in the macula, where high-resolution acuity is required. There are three different types of cones that respond most strongly to three different wavelengths of light, though there is a large overlap. Your visual cortex takes the combined response of the three types of cones and makes up the blended color in your mind. Magenta for example is an imaginary color. It's the color your brain makes up to explain why both short and long wavelengths of light are detected, but not the wavelengths in the middle. Most colors are on the color spectrum you learned in school (Red, Orange, Yellow, Green, Blue, Indigo, Violet), and will trigger a single cone type, or two adjacent cone types.

* '''Retina''' - the tissue that supports the rods and cones.
** '''Macula''' - a small area of the retina with a higher density of light receptors.
** '''Fovea''' - a tiny pit in the macula with the highest density of cones, for highest resolution vision.
* '''Choroid''' - the structure behind the sclera. It can change thickness (on a timescale of days) to make small adjustments to the [[axial length]]
* '''Vitreous humor''' is the clear gel filling the majority of the eyeball. It is where true [[floaters]] live. This gel is important for helping the eye hold its shape and maintain the correct pressures inside the eye even when air pressure changes. In adults, the gel has a complex structure, with different thicknesses in different parts.
* '''Vitreous detachment''' is attached to the outer wall of the eye in multiple locations, but can become separated. When separated from the retina it does not support the retina fully, and puts you at higher risk for [[retinal detachment]]. It can also leave behind a large [[floater]] that impairs vision.

== Axial Length ==
The primary cause of differences in [[refractive state]] is the length of the eye, referred to as axial length, relative to the focusing power. Long eyeballs are associated with myopia, as the natural lens of the eye, even when fully relaxed, focuses light too far forward of the retina.

== [[wikipedia:back of the envelope|Back-of-the-envelope]] calculations ==

We can use [[Optics related math]] and some very approximate numbers to give order-of-magnitude estimates of some of the quanties involved.

To estimate the focusing power of an [[emmetropic eye]], we might take the [[axial length]] as around 2.5cm. For [[distance vision]] (parallel incident light) that number is simply the [[focal length]] of the eye at rest, giving 40 [[Diopters]]. If we take the [[near point]] as about 25cm, that requires an additional 4 dpt of focusing power from the lens.

If we now suppose that myopia is due entirely to elongation (ie the focusing power is unchanged), how much does the axial length need to increase to bring the [[blur horizon]] to 40cm ? With a 40 dpt lens and a source object at 40cm, the image would form 26.67mm from the lens, giving an estimate of elongation of 1.67mm or 6%.

In the same way, we can calculate the new near point : with a lens of 44dpt and an image location of 26.67mm, the source object would be at around 15cm.

==See Also==
* [[Duochrome Test]]

==References==
{{reflist}}

[[Category:Articles]]