Difference between revisions of User:Divenal/sandbox

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A reminder about some trigonometry / geometry

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

# <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>

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

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

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

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

# <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". "20/20", or 1.0, is the ability to resolve features subtending 1 minute of arc.

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:

* 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~~

* On a Landolt C chart, the ~~gaps in~~ the ~~circles, like the 'C's on the Snellen chart, are 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>

~~Acuity~~ <math> = 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"

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

* 1.0 is "20/200"

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

* -0.3 is "20/10"

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

On a Snellen chart, one ~~useful way~~ to ~~think~~ of the ~~labels on~~ the ~~letter rows~~ is as "~~the~~ distance ~~at which they subtend 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.

~~On a~~ chart ~~designed~~ for ~~use at 20 feet, the letters on~~ the "20/50" line ~~are of such a height that they subtend 5 arc-minutes at a~~

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.

~~distance of 50 feet. By the various identities above~~, ~~they are 2.5~~ times ~~the~~ height of the "20/20" ~~letters~~, ~~and they subtend~~

~~an angle of~~ 12.5 arc~~-minutes at 20 feet~~.

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.

@@ Line 50: / Line 50: @@
 A reminder about some trigonometry / geometry
-# <math>tan(A) = \frac{a}{b}</math>
+# <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>
+# for very small angles (expressed in radians), <math>\tan(A) \approx A</math>
-#* so at small angles, 'b' is proportional to 'A'
+#* so at small angles, 'a' is proportional to 'A'
-# 1 radian = <math>\frac{180}{\pi}</math> degrees = <math>\frac{60 \times 180}{\pi}</math> arc-minutes
+# <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". "20/20", or 1.0, is the ability to resolve features subtending 1 minute of arc.
+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:
-* 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
-* On a Landolt C chart, the gaps in the circles, like the 'C's on the Snellen chart, are 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>
-Acuity <math> = 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"
+* 0.0 corresponds to "20/20" (1.0 or 1 arc-minute)
-* 1.0 is "20/200"
+* 1.0 is "20/200" (0.1 or 10 arc-minutes)
-* -0.3 is "20/10"
+* -0.3 is "20/10" (2.0 or 0.5 arc-minutes)
-On a Snellen chart, one useful way to think of the labels on the letter rows is as "the distance at which they subtend 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.
-On a chart designed for use at 20 feet, the letters on the "20/50" line are of such a height that they subtend 5 arc-minutes at a
+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.
-distance of 50 feet. By the various identities above, they are 2.5 times the height of the "20/20" letters, and they subtend
-an angle of 12.5 arc-minutes at 20 feet.
+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.