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'''Binary angular measurement''' ('''BAM''')<ref name="ship"/> (and the '''binary angular measurement system''', '''BAMS'''<ref name="BAMS"/>) is a measure of [[angle]]s using [[binary number]]s and [[fixed-point arithmetic]], in which a full [[turn (angle)|turn]] is represented by the value 1.
These representation of angles are often used in [[numerical control]] and [[digital signal processing]] applications, such as robotics, navigation,<ref name="lap2004"/> computer games,<ref name="sang1993"/> and digital sensors,<ref name="para2005"/> taking advantage of the implicit modular reduction achieved by truncating binary numbers. It may also be used as the [[fractional part]] of a [[fixed-point number]] counting the number of full rotations of e.g. a vehicle's wheels or a [[leadscrew]].
== Representation ==
=== Unsigned fraction of turn ===
In this system, an angle is represented by an ''n''-[[bit]] unsigned binary number in the sequence 0, ..., 2<sup>''n''</sup>−1 that is interpreted as a multiple of 1/2<sup>''n''</sup> of a full turn; that is, 360/2<sup>''n''</sup> degrees or 2π/2<sup>''n''</sup> radians. The number can also be interpreted as a fraction of a full turn between 0 (inclusive) and 1 (exclusive) represented in binary fixed-point format with a scaling factor of 1/2<sup>''n''</sup>. Multiplying that fraction by 360° or 2π gives the angle in [[degree (angle)|degree]]s in the range 0 to 360, or in [[radian]]s, in the range 0 to 2π, respectively.
For example, with ''n'' = 8, the binary integers 00000000<sub>2</sub> (0.00), 01000000<sub>2</sub> (0.25), 10000000<sub>2</sub> (0.50), and 11000000<sub>2</sub> (0.75) represent the angular measures 0°, 90°, 180°, and 270°, respectively.
The main advantage of this system is that the addition or subtraction of the integer numeric values with the ''n''-bit arithmetic used in most computers produces results that are consistent with the geometry of angles. Namely, the integer result of the operation is automatically reduced [[modular arithmetic|modulo]] 2<sup>''n''</sup>, matching the fact that angles that differ by an integer number of full turns are equivalent. Thus one does not need to explicitly test or handle the wrap-around, as one must do when using other representations (such as number of degrees or radians in floating-point).<ref name="harg2019"/>
=== Signed fraction of turn ===
[[Image:Binary angles.svg|360px|thumb|Signed binary angle measurement. {{black|Black}} is traditional degrees representation, {{green|green}} is a BAM as a decimal number and {{red|red}} is [[hexadecimal]] 32-bit BAM. In this figure the 32-bit binary integers are interpreted as signed binary fixed-point values with scaling factor 2<sup>−31</sup>, representing fractions between −1.0 (inclusive) and +1.0 (exclusive).]]
Alternatively, the same ''n'' bits can also be interpreted as a signed integer in the range −2<sup>''n''−1</sup>, ..., 2<sup>''n''−1</sup>−1 in the [[two's complement]] convention. They can also be interpreted as a fraction of a full turn between −0.5 (inclusive) and +0.5 (exclusive) in signed fixed-point format, with the same scaling factor; or a fraction of half-turn between −1.0 (inclusive) and +1.0 (exclusive) with scaling factor 1/2<sup>''n''−1</sup>.
Either way, these numbers can then be interpreted as angles between −180° (inclusive) and +180° (exclusive), with −0.25 meaning −90° and +0.25 meaning +90°. The result of adding or subtracting the numerical values will have the same sign as the result of adding or subtracting angles, once reduced to this range. This interpretation eliminates the need to reduce angles to the range {{closed-closed|−π, +π}} when computing [[trigonometric functions]].
== Example ==
In the orbital data broadcast by the [[Global Positioning System]], angles are encoded using binary angular measurement. In particular, each satellite broadcasts an [[ephemeris]] containing its six Keplerian [[orbital elements]]. Four of these are angles, which are encoded as 32-bit binary angles. In the lower-precision [[GPS signals#Almanac|almanac]] data, 24-bit binary angles are used.<!--Except for inclination, which is specially encoded, a detail considered too esoteric to mention here.-->
== See also ==
* [[Grade (angle)|Grade]], 1/400 of a full turn
* [[Binary scaling]]
* [[CORDIC]], algorithms for trigonometric functions
* [[Constructible polygon]], including all polygons with 2<sup>''n''</sup> sides
== References ==
{{reflist|refs=
<ref name="ship">{{cite web |title=Binary angular measurement |url=http://www.tpub.com/content/fc/14100/css/14100_314.htm |archive-url=https://web.archive.org/web/20091221160257/http://www.tpub.com/content/fc/14100/css/14100_314.htm |archive-date=2009-12-21}}</ref>
<ref name="BAMS">{{cite web |title=Binary Angular Measurement System |work=acronyms.thefreedictionary |url=http://acronyms.thefreedictionary.com/Binary+Angular+Measurement+System}}</ref>
<ref name="harg2019">{{cite web |title=Angles, integers, and modulo arithmetic |author-first=Shawn |author-last=Hargreaves |author-link=:pl:Shawn Hargreaves |publisher=blogs.msdn.com |url=http://blogs.msdn.com/shawnhar/archive/2010/01/04/angles-integers-and-modulo-arithmetic.aspx |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190630223817/http://www.shawnhargreaves.com/blogindex.html |archive-date=2019-06-30}}</ref>
<ref name="lap2004">{{cite book |title=Real-Time Systems Design and Analysis |chapter=Chapter 7.5.3, Binary Angular Measure |author-first=Phillip A. |author-last=LaPlante |date=2004 |publisher=Wiley |chapter-url=http://www.globalspec.com/reference/14722/160210/Chapter-7-5-3-Binary-Angular-Measure |isbn=0-471-22855-9}}</ref>
<ref name="sang1993">{{cite web |title=Doom 1993 code review - Section "Walls" |author-first=Fabien |author-last=Sanglard |date=2010-01-13 |website=fabiensanglard.net |url=http://fabiensanglard.net/doomIphone/doomClassicRenderer.php}}</ref>
<ref name="para2005">{{cite web |title=Hitachi HM55B Compass Module (#29123) |series=Parallax Digital Compass Sensor (#29123) |publisher=[[Parallax, Inc. (company)|Parallax, Inc.]] |date=May 2005 |website=www.hobbyengineering.com |via=www.parallax.com |url=http://www.hobbyengineering.com/specs/PX-29123.pdf |url-status=dead |archive-url=https://web.archive.org/web/20110711172521/http://www.hobbyengineering.com/specs/PX-29123.pdf |archive-date=2011-07-11}}</ref>
}}
[[Category:Units of plane angle]]
[[Category:Binary arithmetic]]
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