Turn (angle)

{{short description|Unit of plane angle where a full circle equals 1}}

{{Infobox unit

| name = Turn

| othernames = Revolution, Cycles

| image = angle-fractions.png

| caption = {{longitem|Counterclockwise rotations about the center point starting from the right, where a complete rotation corresponds to an angle of rotation of 1 turn.}}

| standard =

| quantity = Plane angle

| symbol = tr

| symbol2 = pla

| symbol3 = rev

| symbol4 = cyc

| units1 = radians

| inunits1 = {{math|2π}} rad
≈ {{val|6.283185307|end=...|u=rad}}

| units3 = milliradians

| inunits3 = {{math|2000π}} mrad
≈ {{val|6283.185307|end=...|u=mrad}}

| units4 = degrees

| inunits4 = 360°

| units5 = gradians

| inunits5 = 400^g

}}

The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to {{math|2π}} radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c) or to one revolution (symbol rev or r). Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm).{{efn|The angular unit terms "cycles" and "revolutions" are also used, ambiguously, as shorter versions of the related frequency units.{{cn|date=July 2023}}}} The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers), rotating objects, and the winding number of curves.

Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.

In the ISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless quantity called rotation, defined as the ratio of a given angle and a full turn. It is represented by the symbol N. {{xref|(See below for the formula.)}}

Because one turn is $2\pi$ radians, some have proposed representing $2\pi$ with the single letter tau ( $\tau$ ).

Unit symbols

There are several unit symbols for the turn.

= EU and Switzerland =

The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: {{lang|la|plenus angulus}} 'full angle') for turns. Covered in {{ill|DIN 1301-1|de}} (October 2010), the so-called {{lang|de|Vollwinkel}} ('full angle') is not an SI unit. However, it is a legal unit of measurement in the EU and Switzerland.

= Calculators =

The scientific calculators HP 39gII and HP Prime support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to newRPL for the HP 50g in 2016, and for the hp 39g+, HP 49g+, HP 39gs, and HP 40gs in 2017. An angular mode TURN was suggested for the WP 43S as well, but the calculator instead implements "MUL{{pi}}" (multiples of {{pi}}) as mode and unit since 2019.

Divisions

Many angle units are defined as a division of the turn. For example, the degree is defined such that one turn is 360 degrees.

Using metric prefixes, the turn can be divided in 100 centiturns or {{val|1000}} milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″. A protractor divided in centiturns is normally called a "percentage protractor". While percentage protractors have existed since 1922, the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962. Some measurement devices for artillery and satellite watching carry milliturn scales.

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, which implicitly have an angular separation of {{sfrac|1|32}} turn. The binary degree, also known as the binary radian (or brad), is {{sfrac|1|256}} turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into {{math|2ⁿ}} equal parts for other values of {{mvar|n}}.

Unit conversion

File:2pi-unrolled.gif of the unit circle (whose radius is one) is {{math|2π}}.]]

One turn is equal to $2\pi$ = $\tau$ ≈ {{val|6.283185307179586}} radians, 360 degrees, or 400 gradians.

class="wikitable" style="text-align:center;" \|+ Conversion of common angles
Turns ! colspan="2" \| Radians ! Degrees ! Gradians
0 turn \| colspan="2" \| 0 rad \| 0° \| 0^g
{{sfrac\|1\|72}} turn \| {{sfrac\|{{tau}}\|72}} rad \| {{sfrac\|{{pi}}\|36}} rad \| 5° \| {{sfrac\|5\|5\|9}}^g
{{sfrac\|1\|24}} turn \| {{sfrac\|{{tau}}\|24}} rad \| {{sfrac\|{{pi}}\|12}} rad \| 15° \| {{sfrac\|16\|2\|3}}^g
{{sfrac\|1\|16}} turn \| {{sfrac\|{{tau}}\|16}} rad \| {{sfrac\|{{pi}}\|8}} rad \| 22.5° \| 25^g
{{sfrac\|1\|12}} turn \| {{sfrac\|{{tau}}\|12}} rad \| {{sfrac\|{{pi}}\|6}} rad \| 30° \| {{sfrac\|33\|1\|3}}^g
{{sfrac\|1\|10}} turn \| {{sfrac\|{{tau}}\|10}} rad \| {{sfrac\|{{pi}}\|5}} rad \| 36° \| 40^g
{{sfrac\|1\|8}} turn \| {{sfrac\|{{tau}}\|8}} rad \| {{sfrac\|{{pi}}\|4}} rad \| 45° \| 50^g
{{sfrac\|1\|2{{pi}}}} turn \| colspan="2" \| 1 rad \| {{circa}} 57.3° \| {{circa}} 63.7^g
{{sfrac\|1\|6}} turn \| {{sfrac\|{{tau}}\|6}} rad \| {{sfrac\|{{pi}}\|3}} rad \| 60° \| {{sfrac\|66\|2\|3}}^g
{{sfrac\|1\|5}} turn \| {{sfrac\|{{tau}}\|5}} rad \| {{sfrac\|2{{pi}}\|5}} rad \| 72° \| 80^g
{{sfrac\|1\|4}} turn \| {{sfrac\|{{tau}}\|4}} rad \| {{sfrac\|{{pi}}\|2}} rad \| 90° \| 100^g
{{sfrac\|1\|3}} turn \| {{sfrac\|{{tau}}\|3}} rad \| {{sfrac\|2{{pi}}\|3}} rad \| 120° \| {{sfrac\|133\|1\|3}}^g
{{sfrac\|2\|5}} turn \| {{sfrac\|2{{tau}}\|5}} rad \| {{sfrac\|4{{pi}}\|5}} rad \| 144° \| 160^g
{{sfrac\|1\|2}} turn \| {{sfrac\|{{tau}}\|2}} rad \| {{pi}} rad \| 180° \| 200^g
{{sfrac\|3\|4}} turn \| {{sfrac\|3{{tau}}\|4}} rad \| {{sfrac\|3{{pi}}\|2}} rad \| 270° \| 300^g
1 turn \| {{tau}} rad \| 2{{pi}} rad \| 360° \| 400^g

class="wikitable" style="text-align:center;"

|+ Conversion of common angles

Turns

! colspan="2" | Radians

! Degrees

! Gradians

0 turn

| colspan="2" | 0 rad

| 0°

| 0^g

{{sfrac|1|72}} turn

| {{sfrac|{{tau}}|72}} rad

| {{sfrac|{{pi}}|36}} rad

| 5°

| {{sfrac|5|5|9}}^g

{{sfrac|1|24}} turn

| {{sfrac|{{tau}}|24}} rad

| {{sfrac|{{pi}}|12}} rad

| 15°

| {{sfrac|16|2|3}}^g

{{sfrac|1|16}} turn

| {{sfrac|{{tau}}|16}} rad

| {{sfrac|{{pi}}|8}} rad

| 22.5°

| 25^g

{{sfrac|1|12}} turn

| {{sfrac|{{tau}}|12}} rad

| {{sfrac|{{pi}}|6}} rad

| 30°

| {{sfrac|33|1|3}}^g

{{sfrac|1|10}} turn

| {{sfrac|{{tau}}|10}} rad

| {{sfrac|{{pi}}|5}} rad

| 36°

| 40^g

{{sfrac|1|8}} turn

| {{sfrac|{{tau}}|8}} rad

| {{sfrac|{{pi}}|4}} rad

| 45°

| 50^g

{{sfrac|1|2{{pi}}}} turn

| colspan="2" | 1 rad

| {{circa}} 57.3°

| {{circa}} 63.7^g

{{sfrac|1|6}} turn

| {{sfrac|{{tau}}|6}} rad

| {{sfrac|{{pi}}|3}} rad

| 60°

| {{sfrac|66|2|3}}^g

{{sfrac|1|5}} turn

| {{sfrac|{{tau}}|5}} rad

| {{sfrac|2{{pi}}|5}} rad

| 72°

| 80^g

{{sfrac|1|4}} turn

| {{sfrac|{{tau}}|4}} rad

| {{sfrac|{{pi}}|2}} rad

| 90°

| 100^g

{{sfrac|1|3}} turn

| {{sfrac|{{tau}}|3}} rad

| {{sfrac|2{{pi}}|3}} rad

| 120°

| {{sfrac|133|1|3}}^g

{{sfrac|2|5}} turn

| {{sfrac|2{{tau}}|5}} rad

| {{sfrac|4{{pi}}|5}} rad

| 144°

| 160^g

{{sfrac|1|2}} turn

| {{sfrac|{{tau}}|2}} rad

| {{pi}} rad

| 180°

| 200^g

{{sfrac|3|4}} turn

| {{sfrac|3{{tau}}|4}} rad

| {{sfrac|3{{pi}}|2}} rad

| 270°

| 300^g

1 turn

| {{tau}} rad

| 2{{pi}} rad

| 360°

| 400^g

In the ISQ/SI

{{Infobox physical quantity

| name = Rotation

| othernames = number of revolutions, number of cycles, number of turns, number of rotations

| width =

| background =

| image =

| caption =

| unit = Unitless

| otherunits =

| symbols = N

| baseunits =

| dimension = 1

| extensive =

| intensive =

| conserved =

| transformsas =

| derivations =

}}

In the International System of Quantities (ISQ), rotation (symbol N) is a physical quantity defined as number of revolutions:

N is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by:

: $N = \frac{\varphi}{2 \pi \text{ rad}}$

where {{varphi}} denotes the measure of rotational displacement.

The above definition is part of the ISQ, formalized in the international standard ISO 80000-3 (Space and time), and adopted in the International System of Units (SI).

Rotation count or number of revolutions is a quantity of dimension one, resulting from a ratio of angular displacement.

It can be negative and also greater than 1 in modulus.

The relationship between quantity rotation, N, and unit turns, tr, can be expressed as:

: $N = \frac \varphi \text{tr} = \{ \varphi \}_\text{tr}$

where {{{varphi}}}_tr is the numerical value of the angle {{varphi}} in units of turns (see {{slink|Physical quantity#Components}}).

In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by {{mvar|n}}:

: $n = \frac{\mathrm{d}N}{\mathrm{d}t}$

The SI unit of rotational frequency is the reciprocal second (s⁻¹). Common related units of frequency are hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).

{{Infobox unit

| name = Revolution

| othernames = Cycle

| standard =

| quantity = Rotation

| symbol = rev

| symbol2 = r

| symbol3 = cyc

| symbol4 = c

| units1 = Base units

| inunits1 = 1

}}

The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the dimensionless unit "one",{{efn|"The special name revolution, symbol r, for this unit [name 'one', symbol '1'] is widely used in specifications on rotating machines."{{cite web | title=ISO 80000-3:2006 | website=ISO | date=2001-08-31 | url=https://www.iso.org/standard/31888.html | access-date=2023-04-25}}}}

which also received other special names, such as the radian.{{efn|"Measurement units of quantities of dimension one are numbers. In some cases, these measurement units are given special names, e.g. radian..."}}

Despite their dimensional homogeneity, these two specially named dimensionless units are applicable for non-comparable kinds of quantity: rotation and angle, respectively.{{cite web |title=ISO 80000-1:2009(en) Quantities and units — Part 1: General |url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-1:v1:en |access-date=2023-05-12 |website=iso.org}}

"Cycle" is also mentioned in ISO 80000-3, in the definition of period.{{efn|"3-14) period duration, period: duration (item 3‑9) of one cycle of a periodic event"}}

Notes

References

{{reflist|refs=

{{cite web |title=ooPIC Programmer's Guide - Chapter 15: URCP |work=ooPIC Manual & Technical Specifications - ooPIC Compiler Ver 6.0 |orig-date=1997 |date=2007 |publisher=Savage Innovations, LLC |url=http://www.oopic.com/pgchap15.htm |access-date=2019-08-05 |url-status=dead |archive-url=https://web.archive.org/web/20080628051746/http://www.oopic.com/pgchap15.htm |archive-date=2008-06-28}}

{{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}}

{{cite journal |author-first=Frederick E. |author-last=Croxton |date=1922 |title=A Percentage Protractor - Designed for Use in the Construction of Circle Charts or "Pie Diagrams" |series=Short Note |journal=Journal of the American Statistical Association |volume=18 |issue=137 |pages=108–109 |doi=10.1080/01621459.1922.10502455}}

{{cite book |author-first=Fred |author-last=Hoyle |author-link=Fred Hoyle |editor-first=M. H. |editor-last=Chandler |title=Astronomy |url=https://archive.org/details/astronom00hoyl |url-access=registration |publisher=Macdonald & Co. (Publishers) Ltd. / Rathbone Books Limited |location=London, UK |date=1962 |edition=1 |lccn=62065943 |oclc=7419446}} (320 pages)

{{cite book |author-first=Herbert Arthur |author-last=Klein |title=The Science of Measurement: A Historical Survey (The World of Measurements: Masterpieces, Mysteries and Muddles of Metrology) |chapter=Chapter 8: Keeping Track of Time |edition=corrected reprint of original |date=2012 |orig-date=1988, 1974 |lccn=88-25858 |publisher=Dover Publications, Inc. / Courier Corporation (originally by Simon & Schuster, Inc.) |series=Dover Books on Mathematics |isbn=978-0-48614497-9 |page=102 |chapter-url=https://books.google.com/books?id=CrmuSiCFyikC&pg=PA102 |access-date=2019-08-06}} (736 pages)

{{cite journal |title=Bestimmung von Satellitenbahnen |language=de |author-first=Friedrich |author-last=Schiffner |editor-first=Maria Emma |editor-last=Wähnl |editor-link=:de:Maria Emma Wähnl |journal=Astronomische Mitteilungen der Urania-Sternwarte Wien |publisher=Volksbildungshaus Wiener Urania |location=Wien, Austria |volume=8 |issue= |date=1965}}

{{cite book |title=Trackers of the Skies |author-first=Eugene Nelson |author-last=Hayes |series=History of the Smithsonian Satellite-tracking Program |publisher=Academic Press / Howard A. Doyle Publishing Company |location=Cambridge, Massachusetts, USA |date=1975 |orig-date=1968 |url=https://siris-sihistory.si.edu/ipac20/ipac.jsp?&profile=all&source=~!sichronology&uri=full=3100001~!3190~!0#focus}}

{{cite web |title=Richtlinie 80/181/EWG - Richtlinie des Rates vom 20. Dezember 1979 zur Angleichung der Rechtsvorschriften der Mitgliedstaaten über die Einheiten im Meßwesen und zur Aufhebung der Richtlinie 71/354/EWG |language=de |date=1980-02-15 |url=https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:01980L0181-20090527 |access-date=2019-08-06 |url-status=live |archive-url=https://web.archive.org/web/20190622210052/https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:01980L0181-20090527 |archive-date=2019-06-22}}

{{cite web |title=Richtlinie 2009/3/EG des Europäischen Parlaments und des Rates vom 11. März 2009 zur Änderung der Richtlinie 80/181/EWG des Rates zur Angleichung der Rechtsvorschriften der Mitgliedstaaten über die Einheiten im Messwesen (Text von Bedeutung für den EWR) |language=de |date=2009-03-11 |url=https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:32009L0003&from=EN |access-date=2019-08-06 |url-status=live |archive-url=https://web.archive.org/web/20190806184426/https://eur-lex.europa.eu/legal-content/DE/TXT/PDF/?uri=CELEX:32009L0003&from=EN |archive-date=2019-08-06}}

{{cite book |title=Einheitenverordnung |chapter=Art. 15 Einheiten in Form von nichtdezimalen Vielfachen oder Teilen von SI-Einheiten |id=941.202 |date=1994-11-23 |language=de-ch |publisher=Schweizerischer Bundesrat |chapter-url=http://www.admin.ch/opc/de/classified-compilation/19940345/ |access-date=2013-01-01 |url-status=live |archive-url=https://web.archive.org/web/20190510122902/https://www.admin.ch/opc/de/classified-compilation/19940345/ |archive-date=2019-05-10}}

{{cite book |title=Handbuch SI-Einheiten: Definition, Realisierung, Bewahrung und Weitergabe der SI-Einheiten, Grundlagen der Präzisionsmeßtechnik |author-first1=Sigmar |author-last1=German |author-first2=Peter |author-last2=Drath |publisher=Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, reprint: Springer-Verlag |language=de |date=2013-03-13 |orig-date=1979 |edition=1 |isbn=978-3-32283606-9 |id=978-3-528-08441-7, 978-3-32283606-9 |page=421 |url=https://books.google.com/books?id=63qcBgAAQBAJ&pg=PA421 |access-date=2015-08-14}}

{{cite book |title=Das Vieweg Einheiten-Lexikon: Formeln und Begriffe aus Physik, Chemie und Technik |author-first=Peter |author-last=Kurzweil |language=de |publisher=Vieweg, reprint: Springer-Verlag |edition=1 |date=2013-03-09 |orig-date=1999 |isbn=978-3-32292920-4 |id=978-3-322-92921-1 |doi=10.1007/978-3-322-92920-4 |page=403 |url=https://books.google.com/books?id=2zecBgAAQBAJ |access-date=2015-08-14}}

{{cite web |title=RE: newRPL: Handling of units |author-first=Claudio Daniel |author-last=Lapilli |date=2016-05-11 |work=HP Museum |url=http://www.hpmuseum.org/forum/thread-4783-post-55836.html#pid55836 |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20170810012742/http://www.hpmuseum.org/forum/thread-4783-post-55836.html |archive-date=2017-08-10}}

{{cite book |title=newRPL User Manual |chapter=Chapter 3: Units - Available Units - Angles |author-first=Claudio Daniel |author-last=Lapilli |date=2018-10-25 |chapter-url=https://newrpl.wiki.hpgcc3.org/doku.php?id=manual:chapter3:units#available-units |access-date=2019-08-07 |url-status=live |archive-url=https://web.archive.org/web/20190806225910/https://newrpl.wiki.hpgcc3.org/doku.php?id=manual:chapter3:units#available-units |archive-date=2019-08-06}}

Sequence {{OEIS2C|A019692}}

{{cite web |title=RE: WP-32S in 2016? |date=2016-01-12 |orig-date=2016-01-11 |author-first=Matthias R. |author-last=Paul |work=HP Museum |url=https://www.hpmuseum.org/forum/thread-5427-post-48945.html#pid48945 |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190805163709/https://www.hpmuseum.org/forum/thread-5427-post-48945.html |archive-date=2019-08-05 |quote=[…] I'd like to see a TURN mode being implemented as well. TURN mode works exactly like DEG, RAD and GRAD (including having a full set of angle unit conversion functions like on the WP 34S), except for that a full circle doesn't equal 360 degree, 6.2831... rad or 400 gon, but 1 turn. (I […] found it to be really convenient in engineering/programming, where you often have to convert to/from other unit representations […] But I think it can also be useful for educational purposes. […]) Having the angle of a full circle normalized to 1 allows for easier conversions to/from a whole bunch of other angle units […]}}

{{cite book |title=WP 43S Owner's Manual |date=2019 |orig-date=2015 |author-last=Bonin |author-first=Walter |isbn=978-1-72950098-9 |edition=draft |version=0.12 |pages=72, 118–119, 311 |url=https://gitlab.com/wpcalculators/wp43/-/raw/master/docs/OwnersManual.pdf |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20230718192232/https://gitlab.com/rpncalculators/wp43/-/raw/master/docs/OwnersManual.pdf |archive-date=2023-07-18}} [https://gitlab.com/Over_score/wp43s] [https://gitlab.com/wpcalculators/wp43] (314 pages)

{{cite book |title=WP 43S Reference Manual |date=2019 |orig-date=2015 |author-last=Bonin |author-first=Walter |isbn=978-1-72950106-1 |edition=draft |version=0.12 |pages=iii, 54, 97, 128, 144, 193, 195 |url=https://gitlab.com/wpcalculators/wp43/-/raw/master/docs/ReferenceManual.pdf |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20230718192332/https://gitlab.com/rpncalculators/wp43/-/raw/master/docs/ReferenceManual.pdf |archive-date=2023-07-18}} [https://gitlab.com/Over_score/wp43s] [https://gitlab.com/wpcalculators/wp43] (271 pages)

{{cite book |author-last=Fitzpatrick |author-first=Richard |title=Newtonian Dynamics: An Introduction |publisher=CRC Press |date=2021 |isbn=978-1-000-50953-3 |url=https://books.google.com/books?id=rRpSEAAAQBAJ&pg=PA116 |access-date=2023-04-25 |page=116}}

{{cite book |title=Units & Symbols for Electrical & Electronic Engineers |date=2016 |publisher=Institution of Engineering and Technology |publication-place=London, UK |url=https://www.theiet.org/media/4173/units-and-symbols.pdf |access-date=2023-07-18 |url-status=live |archive-url=https://web.archive.org/web/20230718183635/https://www.theiet.org/media/4173/units-and-symbols.pdf |archive-date=2023-07-18}} (1+iii+32+1 pages)

{{cite web |title=ISO 80000-3:2019 Quantities and units — Part 3: Space and time |publisher=International Organization for Standardization |date=2019 |edition=2 |url=https://www.iso.org/standard/64974.html |access-date=2019-10-23}} [https://www.iso.org/obp/ui/#iso:std:iso:80000:-3:ed-2:v1:en] (11 pages)

{{cite web |title=The NIST Guide for the Use of the International System of Units, Special Publication 811 |author-first1=Ambler |author-last1=Thompson |author-first2=Barry N. |author-last2=Taylor |edition=2008 |publisher=National Institute of Standards and Technology |date=2020-03-04 |orig-date=2009-07-02 |ref={{sfnref|NIST|2009}} |url=https://www.nist.gov/pml/special-publication-811 |access-date=2023-07-17}} [https://web.archive.org/web/20230515201622/https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication811e2008.pdf]

}}