forked from mpusz/mp-units
146 lines
7.2 KiB
ReStructuredText
146 lines
7.2 KiB
ReStructuredText
.. namespace:: units
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Angular Units
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=============
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Some background information
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---------------------------
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As per today's SI, both radian and steradian are dimensionless. This forces the convention to set the angle ``1 radian``
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equal to the number ``1`` within equations (similar to what natural units system does for ``c`` constant).
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Following `Wikipedia <https://en.wikipedia.org/wiki/Radian#Dimensional_analysis>`_:
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Giacomo Prando says "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian
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in the dimensional analysis of physical equations." For example, a mass hanging by a string from a pulley will rise or drop
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by :math:`y=rθ`` centimeters, where :math:`r` is the radius of the pulley in centimeters and :math:`θ` is the angle
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the pulley turns in radians. When multiplying :math:`r` by :math:`θ` the unit of radians disappears from the result.
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Similarly in the formula for the angular velocity of a rolling wheel, :math:`ω=v/r`, radians appear in the units of
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:math:`ω` but not on the right hand side.
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Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical
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advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention
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and contextual knowledge is "pedagogically unsatisfying".
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At least a dozen scientists have made proposals to treat the radian as a base unit of measure defining its own dimension of "angle",
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as early as 1936 and as recently as 2022. This would bring the advantages of a physics-based, consistent, and logically-robust
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unit system, with unambiguous units for all physical quantities. At the same time the only notable changes for typical
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end-users would be: improved units for the quantities **torque**, **angular momentum** and **moment of inertia**.
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Paul Quincey in his proposal *"Angles in the SI: a detailed proposal for solving the problem"* [Quincey]_ states:
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The familiar units assigned to some angular quantities are based on equations that have adopted the radian convention,
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and so are missing `rad`s that would be present if the complete equation is used. The physically-correct units are
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those with the `rad`s reinstated. Numerical values would not change, and the physical meanings of all quantities would
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also be unaffected.
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He proposes the following changes:
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- The **radian** would become either a new base unit or a 'complementary' unit
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- The **steradian** would become a derived unit equal to :math:`1\:rad^2`
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- The SI units for
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- **Torque** would change from :math:`N\:m` (= J) to :math:`J\:rad^{-1}`
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- **Angular momentum** would change from :math:`J\:s` to :math:`J\:s\:rad^{-1}` (i.e. :math:`J/(rad/s)`)
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- **Moment of inertia** would change from :math:`kg\:m^2` to :math:`kg\:m^2\:rad^{-2}` (i.e. :math:`J/(rad/s)^2`)
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- The option to omit the radian from the SI units for **angle**, **angular velocity**, **angular frequency**,
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**angular acceleration**, and **angular wavenumber** would be removed, the only correct SI units being
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:math:`rad`, :math:`rad/s`, :math:`rad/s`, :math:`rad/s^2` and :math:`rad/m` respectively.
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Paul Quincey summarizes that with the above in action:
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However, the physical clarity this would build into the SI should be recognised very quickly. The units would tell us that
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:math:`torque \times angle = energy`, and :math:`angular\:momentum \times angle = action`, for example, in the same way that they do for
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:math:`force \times distance = energy`, :math:`linear\:momentum \times distance = action`, and
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:math:`radiant\:intensity \times solid\:angle = radiant\:flux`.
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Dimensional analysis could be used to its full extent. Software involving angular quantities would be rationalised.
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Arguments about the correct units for frequency and angular frequency, and the meaning of the unit Hz, could be left behind.
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The explanation of these changes would be considerably easier and more rewarding than explaining how a kilogram-sized mass
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can be measured in terms of the Planck constant.
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Angular quantities in the library
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---------------------------------
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The mp-units library strives to define physically-correct quantities and their units to provide maximum help
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to its users. As treating angle as a dimensional quantity can lead to many "trivial" mistakes in dimensional
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analysis and calculation, it was decided to follow the above approach and treat angle as a base quantity
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with a base unit of radian and solid angle as its derived quantity.
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As those the above (at least for now) not a part of :term:`SI`, the plain angle and solid
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angle definitions can be found in the *units/generic* subdirectory. However, in the sake of correctness,
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they are consistently used in the dimensional recipes of all angle-based :term:`SI` quantities like
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torque or angular velocity::
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#include <units/generic/angle.h>
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#include <units/isq/si/torque.h>
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#include <units/math.h>
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#include <units/quantity_io.h>
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#include <iostream>
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int main()
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{
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using namespace units;
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using namespace units::isq;
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using namespace units::aliases;
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using namespace units::aliases::isq::si;
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const Length auto lever = cm<>(20);
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const Force auto force = N<>(500);
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const Angle auto angle = deg<>(90);
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const Torque auto torque = lever * force * sin(angle) / cotes_angle<>;
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std::cout << "Applying a perpendicular force of " << force << " to a "
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<< lever << " long lever results in "
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<< quantity_cast<si::newton_metre_per_radian>(torque) << " of torque.\n";
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}
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The above program prints:
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.. code-block::
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Applying a perpendicular force of 500 N to a 20 cm long lever results in 100 N⋅m/rad of torque.
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.. note::
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`cotes_angle` is a constant which represents an angle with the value of exactly ``1 radian``.
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You can find more information about this constant in [Quincey]_.
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Radians and degrees support
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---------------------------
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Thanks to :ref:`framework/magnitudes:Magnitudes` usage the library provides efficient strong types for all
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angular types. This means that with the built-in support for magnitudes of :math:`\pi` we can provide
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accurate conversions between radians and degrees. The library also provides common trigonometric functions
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for angular quantities::
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using namespace units::aliases::isq::si;
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auto velocity = km_per_h<>(110);
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auto rate_of_climb = m_per_s<>(-0.63657);
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auto glide_ratio = velocity / -rate_of_climb;
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auto glide_angle = asin(1 / glide_ratio);
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std::cout << std::format("Glide ratio: {:%.1Q %q}\n", quantity_cast<units::one>(glide_ratio));
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std::cout << "Glide angle:\n";
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std::cout << std::format(" - {:%.4Q %q}\n", glide_angle);
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std::cout << std::format(" - {:%.2Q %q}\n", quantity_cast<units::degree>(glide_angle));
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std::cout << std::format(" - {:%.2Q %q}\n", quantity_cast<units::gradian>(glide_angle));
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The above program prints:
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.. code-block::
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Glide ratio: 48.0
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Glide angle:
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- 0.0208 rad
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- 1.19 °
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- 1.33 ᵍ
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.. note::
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The above ``velocity`` and ``rate_of_climb`` in the production code should be modelled as
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:ref:`framework/quantity_kinds:Quantity Kinds`.
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A better but longer code can be found in the :ref:`examples/basics/glide_computer:glide_computer` example.
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