Turn (geometry)
Turn | |
---|---|
Unit of | Plane angle |
Symbol | tr or pla |
Conversions | |
1 tr in ... | ... is equal to ... |
radians | 6.283185307179586... rad |
radians | 2π rad |
degrees | 360° |
gradians | 400^{g} |
A turn is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is also referred to as a cycle (abbreviated cyc), revolution (abbreviated rev), complete rotation (abbreviated rot) or full circle.
Subdivisions of a turn include half turns, quarter turns, centiturns, milliturns, points, etc.
Subdivision of turns
A turn can be divided in 100 centiturns or 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.
Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points. The binary degree, also known as the binary radian (or brad), is ^{1}⁄_{256} turn.^{[1]} 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 2^{n} equal parts for other values of n.^{[2]}
The notion of turn is commonly used for planar rotations.
History
The word turn originates via Latin and French from the Greek word τόρνος (tórnos – a lathe).
In 1697, David Gregory used π/ρ (pi over rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius.^{[3]}^{[4]} However, earlier in 1647, William Oughtred had used δ/π (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol π on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the Welsh mathematician William Jones.^{[5]} Euler adopted the symbol with that meaning in 1737, leading to its widespread use.
Percentage protractors have existed since 1922,^{[6]} but the terms centiturns and milliturns were introduced much later by Fred Hoyle.^{[7]}
The German standard DIN 1315 (1974-03) proposed the unit symbol pla (from Latin: plenus angulus "full angle") for turns.^{[8]}^{[9]} Since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was also added to newRPL for the HP 50g.^{[10]} In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.^{[11]}
The standard ISO 80000-3:2006 mentions that the unit name revolution with symbol r is used with rotating machines, as well as using the term turn to mean a full rotation. The standard IEEE 260.1:2004 also uses the unit name rotation and symbol r.
Unit conversion
One turn is equal to 2π (≈ 6.283185307179586)^{[12]} radians.
Turns | Radians | Degrees | Gradians (Gons) |
---|---|---|---|
0 | 0 | 0° | 0^{g} |
1/24 | π/12 | 15° | 16 2/3^{g} |
1/12 | π/6 | 30° | 33 1/3^{g} |
1/10 | π/5 | 36° | 40^{g} |
1/8 | π/4 | 45° | 50^{g} |
1/2π | 1 | c. 57.3° | c. 63.7^{g} |
1/6 | π/3 | 60° | 66 2/3^{g} |
1/5 | 2π/5 | 72° | 80^{g} |
1/4 | π/2 | 90° | 100^{g} |
1/3 | 2π/3 | 120° | 133 1/3^{g} |
2/5 | 4π/5 | 144° | 160^{g} |
1/2 | π | 180° | 200^{g} |
3/4 | 3π/2 | 270° | 300^{g} |
1 | 2π | 360° | 400^{g} |
Tau proposals
In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of π, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive. His proposal used a "pi with three legs" symbol to denote the constant ( = 2π).^{[13]}
In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant: τ = 2π. He offered two reasons. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed more directly: for instance, a 3/4 turn would be represented as 3/4τ rad instead of 3/2π rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable.^{[14]} Hartl's Tau Manifesto^{[15]} gives many examples of formulas that are asserted to be clearer where tau is used instead of pi.^{[16]}^{[17]}^{[18]}
The τ-functionality is made available in the Google calculator and in several programming languages like Python^{[19]}, Perl^{[20]}, Processing^{[21]}, and Nim^{[22]}. It has also been used in at least one mathematical research article,^{[23]} authored by the τ-promoter P. Harremoës.^{[24]}
However, none of these proposals have received widespread acceptance by the mathematical and scientific communities.^{[25]}
Examples of use
- As an angular unit, the turn or revolution is particularly useful for large angles, such as in connection with electromagnetic coils and rotating objects. See also winding number.
- The angular speed of rotating machinery, such as automobile engines, is commonly measured in revolutions per minute or RPM.
- Turn is used in complex dynamics for measure of external and internal angles. The sum of external angles of a polygon equals one turn. Angle doubling map is used.
- Pie charts illustrate proportions of a whole as fractions of a turn. Each one percent is shown as an angle of one centiturn.
Kinematics of turns
In kinematics, a turn is a rotation less than a full revolution. A turn may be represented in a mathematical model that uses expressions of complex numbers or quaternions. In the complex plane every non-zero number has a polar coordinate expression z = r cis(a) = r cos(a) + ri sin(a) where r > 0 and a is in [0, 2π). A turn of the complex plane arises from multiplying z = x + iy by an element u = e^{bi} that lies on the unit circle:
- z ↦ uz.
Frank Morley consistently referred to elements of the unit circle as turns in the book Inversive Geometry, (1933) which he coauthored with his son Frank Vigor Morley.^{[26]}
The Latin term for turn is versor, which is a quaternion that can be visualized as an arc of a great circle. The product of two versors can be compared to a spherical triangle where two sides add to the third. For the kinematics of rotation in three dimensions, see quaternions and spatial rotation.
See also
- Angle of rotation
- Revolutions per minute
- Repeating circle
- Spat (unit) — the 3D counterpart of the turn, equivalent to 4π steradians.
- Unit interval
- Turn (rational trigonometry)
- Spread (rational trigonometry)
- Modulo operation
Other Languages
Copyright
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