mp-units/docs/users_guide/systems/strong_angular_system.md

Strong Angular System

Some background information

As per today's SI, both radian and steradian are dimensionless. This forces the convention to set the angle 1 radian equal to the number 1 within equations (similar to what natural units system does for c constant).

Following Wikipedia:

!!! quote "Wikipedia: Radian - Dimensional analysis"

Giacomo Prando says "the current state of affairs leads inevitably to ghostly appearances
and disappearances of the radian in the dimensional analysis of physical equations."
For example, a mass hanging by a string from a pulley will rise or drop by $y=rθ$
centimeters, where $r$ is the radius of the pulley in centimeters and $θ$ is the
angle the pulley turns in radians. When multiplying $r$ by $θ$ the unit of radians
disappears from the result. Similarly in the formula for the angular velocity of
a rolling wheel, $ω=v/r$, radians appear in the units of $ω$ but not on the right
hand side. Anthony French calls this phenomenon "a perennial problem in the teaching
of mechanics". Oberhofer says that the typical advice of ignoring radians during
dimensional analysis and adding or removing radians in units according to convention
and contextual knowledge is "pedagogically unsatisfying".

At least a dozen scientists have made proposals to treat the radian as a base unit of measure defining its own dimension of "angle", as early as 1936 and as recently as 2022. This would bring the advantages of a physics-based, consistent, and logically-robust unit system, with unambiguous units for all physical quantities. At the same time the only notable changes for typical end-users would be: improved units for the quantities torque, angular momentum, and moment of inertia.

Paul Quincey in his proposal "Angles in the SI: a detailed proposal for solving the problem" states:

!!! quote "Paul Quincey: Angles in the SI: a detailed proposal for solving the problem"

The familiar units assigned to some angular quantities are based on equations that
have adopted the radian convention, and so are missing `rad`s that would be present
if the complete equation is used. The physically-correct units are those with the
`rad`s reinstated. Numerical values would not change, and the physical meanings of
all quantities would also be unaffected.

He proposes the following changes:

• The radian would become either a new base unit or a 'complementary' unit

• The steradian would become a derived unit equal to 1\:rad^2

• The SI units for

  • Torque would change from N\:m (=J) to J\:rad^{-1}
  • Angular momentum would change from J\:s to J\:s\:rad^{-1} (i.e. J/(rad/s))
  • Moment of inertia would change from kg\:m^2 to kg\:m^2\:rad^{-2} (i.e. J/(rad/s)^2)

• The option to omit the radian from the SI units for angle, angular velocity, angular frequency, angular acceleration, and angular wavenumber would be removed, the only correct SI units being rad, rad/s, rad/s, rad/s^2 and rad/m respectively.

Paul Quincey summarizes that with the above in action:

!!! quote "Paul Quincey: Angles in the SI: a detailed proposal for solving the problem"

However, the physical clarity this would build into the SI should be recognised
very quickly. The units would tell us that $torque \times angle = energy$, and
$angular\:momentum \times angle = action$, for example, in the same way that they
do for $force \times distance = energy$, $linear\:momentum \times distance = action$,
and $radiant\:intensity \times solid\:angle = radiant\:flux$.
Dimensional analysis could be used to its full extent. Software involving angular
quantities would be rationalised. Arguments about the correct units for frequency
and angular frequency, and the meaning of the unit $Hz$, could be left behind.
The explanation of these changes would be considerably easier and more rewarding
than explaining how a kilogram-sized mass can be measured in terms of the Planck
constant.

Angular quantities in the SI

Even though the SI somehow ignores the dimensionality of angle:

!!! quote "SI Brochure"

Plane and solid angles, when expressed in radians and steradians respectively, are
in effect also treated within the SI as quantities with the unit one. The symbols
$rad$ and $sr$ are written explicitly where appropriate, in order to emphasize that,
for radians or steradians, the quantity being considered is, or involves the plane
angle or solid angle respectively. For steradians it emphasizes the distinction
between units of flux and intensity in radiometry and photometry for example.
However, it is a long-established practice in mathematics and across all areas of
science to make use of $rad = 1$ and $sr = 1$. For historical reasons the radian
and steradian are treated as derived units.

It also explicitly states:

!!! quote "SI Brochure"

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 factor of $2\pi$. Ignoring this fact may cause an error of $2\pi$.
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 $s^{-1}$.

Strong Angular extensions in the library

The mp-units library strives to define physically-correct quantities and their units to provide maximum help to its users. As treating angle as a dimensional quantity can lead to many "trivial" mistakes in dimensional analysis and calculation, it was decided to provide additional experimental systems of quantities and units that follow the above approach and treat angle as a base quantity with a base unit of radian and solid angle as its derived quantity.

The library provides two related but distinct Strong Angular Systems:

The angular System

This is a standalone system that provides:

• A strong dimension for angle (distinct from dimensionless)
• Base quantities: angle and derived solid angle
• Units: radian (rad), degree (deg), and other angular units
• Trigonometric functions working with angular quantities

Use this system when you want explicit angular dimensions but don't need the full ISQ quantity hierarchy.

The isq_angle System

This system amends the International System of Quantities (ISQ) by incorporating the strong angular definitions from the angular system. It redefines angle-based ISQ quantities to make their recipes physically correct:

• Uses angular::angle as the base for ISQ angular quantities
• Provides correct dimensional analysis for quantities like torque, angular velocity, and angular momentum
• Maintains compatibility with the ISQ quantity hierarchy

Use this system when you need both strong angular dimensions and the full typed ISQ quantity system.

Example

Here's an example using isq_angle to calculate torque with proper dimensional analysis:

using namespace mp_units;
using namespace mp_units::si::unit_symbols;
using mp_units::angular::unit_symbols::deg;
using mp_units::angular::unit_symbols::rad;

const quantity lever = isq_angle::position_vector(20 * cm);
const quantity force = isq_angle::force(500 * N);
const quantity angle = isq_angle::angular_measure(90. * deg);
const quantity torque = isq_angle::torque(lever * force * angular::sin(angle) / (1 * isq_angle::cotes_angle));

std::cout << "Applying a perpendicular force of " << force << " to a " << lever << " long lever results in "
          << torque.in(N * m / rad) << " of torque.\n";

The above program prints:

Applying a perpendicular force of 500 N to a 20 cm long lever results in 100 N m/rad of torque.

!!! note

`cotes_angle` is a constant which represents an angle with the value of exactly `1 radian`.
You can find more information about this constant in [Quincey](../../reference/bibliography.md#Quincey).

Try it live on Compiler Explorer{ .md-button }

References

• Angular Systems Reference - Complete list of quantities and units
• ISQ Angle Systems Reference - ISQ with strong angular dimensions
• Quincey: Angles in the SI - Detailed proposal for solving the angle problem