Binary angular measurement: Difference between revisions

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The terms '''binary angular measurement''' ('''BAM''')<ref name="ship"/> and '''binary angular measurement system''' ('''BAMS''')<ref name="BAMS"/> refer to certain methodologies for representing and manipulating [[angle]]s using [[binary number|binary]] ([[number base|base]] 2) [[fixed-point arithmetic]]. The [[unit of measure|unit]] of angular measure used in those methods may be called '''binary radian''' ('''brad''') or '''binary degree'''.

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"/> On the other hand, this system is not adequate for situations where the number of full turns[[turn (angle)|turn]]s must be measured, e.g. to monitor the rotation of vehicle wheels or of [[leadscrew]]s.

[[Image:Binary angles.svg|360px|thumb|Binary angle measurement system. <span style="color:black">Black</span> is traditional degrees representation, <span style="color:green">green</span> is a BAM as a decimal number and <span style="color:red">red</span> is [[hexadecimal]] 32&nbsp;-bit BAM. In this figure the 32-bit binary integers are interpreted as signed binary fixed-point values with scaling factor 2⁻³¹, representing fractions between −1.0 (inclusive) and +1.0 (exclusive).]]

==Representation==

===Unsigned fraction of turn===

In this system, an angle is represented by an ''n''-[[bit]] unsigned binary number in the sequence 0, ..., 2^''n''−1 that is interpreted as an multiple of 1/2^''n'' of a full turn; that is, 360/2^''n'' degrees or 2π/2^''n'' 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^''n''. 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)₂ (fraction 0.00), (01000000)₂ (0.25), (10000000)₂ (0.50), and (11000000)₂ (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^''n'', 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===

Alternatively, the same ''n'' bits can also be interpreted as a signed integer in the range −2^''n''−1, ..., 2^''n''−1−1 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^''n''−1.

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==See also==

* [[Turn (angle)|Turn]]

* [[Grade (angle)|Grade]], 1/400 of a full turn.

* [[CORDIC]], algorithms for trigonometric functions.

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==References==

{{reflist|refs=

<references>

<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=harg2019"lap2004">{{cite webbook |title=Angles,Real-Time integers,Systems Design and moduloAnalysis arithmetic|chapter=Chapter 7.5.3, Binary Angular Measure |author-first=ShawnPhillip A. |author-last=HargreavesLaPlante |author-linkdate=:pl:Shawn Hargreaves2004 |publisherwebsite=blogswww.msdnglobalspec.com |chapter-url=http://blogswww.msdnglobalspec.com/shawnharreference/archive14722/2010160210/01/04/anglesChapter-integers7-and5-modulo3-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=2019Binary-06Angular-30Measure}}</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=lap2004"para2005">{{cite bookweb |title=Real-TimeHitachi SystemsHM55B DesignCompass andModule Analysis(#29123) |chapterseries=ChapterParallax 7.5.3,Digital BinaryCompass AngularSensor Measure(#29123) |author-firstpublisher=Phillip[[Parallax, AInc. (company)|author-last=LaPlanteParallax, Inc.]] |date=2004May 2005 |website=www.globalspechobbyengineering.com |via=www.parallax.com |chapter-url=http://www.globalspechobbyengineering.com/referencespecs/14722/160210/ChapterPX-729123.pdf |url-5status=dead |archive-3url=https://web.archive.org/web/20110711172521/http://www.hobbyengineering.com/specs/PX-Binary29123.pdf |archive-date=2011-Angular07-Measure11}}</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>

</references>

[[Category:Units of plane angle]]