Difference between revisions of Optics related math

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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==

@@ Line 94: / Line 94: @@
-==Visual acuity equation==
+==Visual acuity==
-<math>(\frac{font\ height}{distance\ to\ sign})(\frac{180}{pi}) \times 60 = arcminutes = a</math>
+[[File:TrigonometryTriangle.svg|right|TrigonometryTriangle]]
-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==