User:Fgnievinski/Angles and related quantities in the International System of Quantities (draft)

The International System of Quantities (ISQ) defines several quantities related to to angles in the international standard ISO 80000-3:
 * phase angle or just phase (φ): the argument of a complex number; normally in radians (bounded to 0-2π);
 * angular measure or just angle (θ): measure of a plane angle figure; normally in radians (0-2π);
 * angular displacement (also θ): ratio between the traversed circular arc length and the circle radius; normally in radins, with sign to indicate clockwise or counterclockwise displacement (doubly unbounded: -∞;+∞); agrees with angular measure within a full-circle domain, 0-2π;
 * rotation (N): the number of revolutions (not necessarily integer); also known as rotation number or turning number; unitless, although common names for the unit one include "cycles" (cyc), "revolutions" (rev), and "turns" (tr); it is a normalized angular value equal to either the ratio of an angular measure and the full circle, N=θ/2π (bounded to 0-1), or the ratio of angular displacement and the full circle, N=Θ/2π (doubly unbounded: -∞;+∞);

The corresponding temporal rates include:
 * angular frequency or angular speed (ω): rate of change of the phase angle (or angular displacement) with respect to time, ω=dφ/dt or ω=dΘ/dt; in rad/s (radians per second).
 * rotational frequency (n): rate of change of rotation, n=dN/dt; in s^-1 (reciprocal second).

A vector quantity related to the above scalar quantities is:
 * angular velocity: vector with magnitude equal to angular speed and direction defining an axis of rotation

Notes:
 * rotational quantities and angular quantities are different kind of quantities -- not unlike amount of substance (in moles) and number of particles (unitless).
 * rotational displacement should be a synonym of rotation; however, sometimes it is mis-associated with angular displacement;
 * the term "number of revolutions" is recognized (ISQ) as a synonym for the quantity rotations. Consequently, the term "revolutions" could be used to refer to the corresponding unitless unit. It cannot be used as a unit for angular quantities. For example, one may say the number of revolutions equals five or the rotation equals five revolutions. But one may not say the angle equals five revolutions. By extension, the term "revolutions per second" is a coherently defined synonym for the unit reciprocal seconds when applied to rotational frequency quantities.
 * the term "cycles per second" is recognized (SI Brochure) as a synonym for the units of reciprocal seconds when applied to rotational frequency quantities; it may not be used to refer to angular frequency quantities. By logical extension, the term "cycles" is a coherent synonym for the unitless quantity rotation. So one may say the rotation equals five or five cycles. It may not be used for angular quantities.
 * units alone are not sufficient in conveying the measurement being reported, the kind of quantity is also essential in reporting; despite the radian being equivalent to one, angular units and rotational units are not to be confused; for the same reason, angular frequency units (rad/s=s^-1) and rotational frequency units (s^-1) are also not to be confused.

Quotes:
 * SI Brocure (9th ed.): "The SI unit of frequency is hertz, the SI unit of angular velocity and angular frequency is radian per second, and the SI unit of activity is becquerel, implying counts per second. Although it is formally correct to write all three of these units as the reciprocal second, the use of the different names emphasizes the different nature of the quantities concerned. It is especially important to carefully distinguish frequencies from angular frequencies, because by definition their numerical values differ by a factor1 of 2π. Ignoring this fact may cause an error of 2π. Note that in some countries, frequency values are conventionally expressed using “cycle/s” or “cps” instead of the SI unit Hz, although “cycle” and “cps” are not units in the SI. Note also that it is common, although not recommended, to use the term frequency for quantities expressed in rad/s. Because of this, it is recommended that quantities called “frequency”, “angular frequency”, and “angular velocity” always be given explicit units of Hz or rad/s and not s^−1."
 * NIST Guide to the SI: "The rotational frequency n of a rotating body is defined to be the number of revolutions it makes in a time interval divided by that time interval [4: ISO 80000-3]. The SI unit of this quantity is thus the reciprocal second (s^−1). However, as pointed out in Ref. [4: ISO 80000-3], the designations “revolutions per second” (r/s) and “revolutions per minute” (r/min) are widely used as units for rotational frequency in specifications on rotating machinery."