Optics related math - Revision history

Angel at 08:20, 28 January 2022

2022-01-28T08:20:14Z

← Older revision		Revision as of 08:20, 28 January 2022
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	* [[Vertex distance]]		* [[Vertex distance]]
	* [[Wikipedia:Vertex_distance\|Vertex distance formula (also for astigmatism)]]		* [[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==		==References==
	{{reflist}}		{{reflist}}

NottNott: Undo revision 15958 by NottNott (talk)

2022-01-26T04:28:50Z

Undo revision 15958 by NottNott (talk)

← Older revision		Revision as of 04:28, 26 January 2022
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	[[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.		Here's a page with maths related to diopters and glasses.

NottNott: Undo revision 15350 by NottNott (talk)

2022-01-26T04:28:02Z

Undo revision 15350 by NottNott (talk)

← Older revision		Revision as of 04:28, 26 January 2022
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	~~{{gif fixer\|~~[[File:Math guy with glasses.gif]]~~\|right}}~~		[[File:Math guy with glasses.gif\|right]]
	Here's a page with maths related to diopters and glasses.		Here's a page with maths related to diopters and glasses.

NottNott: accessible gif

2021-10-17T01:11:03Z

accessible gif

← Older revision		Revision as of 01:11, 17 October 2021
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	[[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.		Here's a page with maths related to diopters and glasses.

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

2020-08-05T15:26:00Z

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

← Older revision		Revision as of 15:26, 5 August 2020
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	Remember that 100cm = 1m.		Remember that 100cm = 1m.

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

	conversely		conversely

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

	==Calculating correction==		==Calculating correction==

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

2020-08-05T15:24:44Z

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

← Older revision		Revision as of 15:24, 5 August 2020
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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==		==Average axial length accomodation/rate of change==

Divenal: /* Diverging lens */ add alternative viewpoint of how -ve lens work

2020-07-30T19:38:09Z

Diverging lens: add alternative viewpoint of how -ve lens work

← Older revision		Revision as of 19:38, 30 July 2020
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	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.		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===		===Diopters are inverse meters===

Divenal: /* Average axial length accomodation/rate of change */ don't try to write sentences in math mode.

2020-07-30T16:03:50Z

Average axial length accomodation/rate of change: don't try to write sentences in math mode.

← Older revision		Revision as of 16:03, 30 July 2020
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	==Average axial length accomodation/rate of change==		==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		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		about 0.92microns/day - 0.69microns/day average
	''Credit: [https://www.facebook.com/groups/endmyopia/permalink/759426960917375/ Mark Podowski]''		''Credit: [https://www.facebook.com/groups/endmyopia/permalink/759426960917375/ Mark Podowski]''

	==Converting from Glasses to Contact Lens Prescription or vice-versa==		==Converting from Glasses to Contact Lens Prescription or vice-versa==
	* [[Vertex distance]]		* [[Vertex distance]]

Divenal: add a new subsection

2020-07-30T15:02:20Z

add a new subsection

← Older revision		Revision as of 15:02, 30 July 2020
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	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.		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.		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.

Divenal: /* Infinity */ equivalent to virtual image at - infty

2020-07-30T15:01:17Z

Infinity: equivalent to virtual image at - infty

← Older revision		Revision as of 15:01, 30 July 2020
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	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.		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.		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===		===Virtual image===