docs: Angular Units chapter added

docs/CMakeLists.txt

@@ -82,6 +82,7 @@ set(unitsSphinxDocs
     "${CMAKE_CURRENT_SOURCE_DIR}/examples/kalman_filter/kalman.rst"
     "${CMAKE_CURRENT_SOURCE_DIR}/faq.rst"
     "${CMAKE_CURRENT_SOURCE_DIR}/framework.rst"
+    "${CMAKE_CURRENT_SOURCE_DIR}/framework/angular_units.rst"
     "${CMAKE_CURRENT_SOURCE_DIR}/framework/arithmetics.rst"
     "${CMAKE_CURRENT_SOURCE_DIR}/framework/basic_concepts.rst"
     "${CMAKE_CURRENT_SOURCE_DIR}/framework/constants.rst"

docs/framework.rst

@@ -22,4 +22,5 @@ Framework Basics
     framework/arithmetics
     framework/constants
     framework/conversions_and_casting
+    framework/angular_units
     framework/text_output

docs/framework/angular_units.rst

@@ -0,0 +1,100 @@
+.. namespace:: units
+
+Angular Units
+=============
+
+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 <https://en.wikipedia.org/wiki/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 :math:`y=rθ`` centimeters, where :math:`r` is the radius of the pulley in centimeters and :math:`θ` is the angle
+    the pulley turns in radians. When multiplying :math:`r` by :math:`θ` the unit of radians disappears from the result.
+    Similarly in the formula for the angular velocity of a rolling wheel, :math:`ω=v/r`, radians appear in the units of
+    :math:`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" <https://arxiv.org/pdf/2108.05704.pdf>`_ states:
+
+    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 :math:`1\:rad^2`
+- The SI units for
+  
+  - **Torque** would change from :math:`N\:m` (= J) to :math:`J\:rad^{-1}`
+  - **Angular momentum** would change from :math:`J\:s` to :math:`J\:s\:rad^{-1}` (i.e. :math:`J/(rad/s)`)
+  - **Moment of inertia** would change from :math:`kg\:m^2` to :math:`kg\:m^2\:rad^{-2}` (i.e. :math:`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
+  :math:`rad`, :math:`rad/s`, :math:`rad/s`, :math:`rad/s^2` and :math:`rad/m` respectively.
+
+
+Angular quantities 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 follow the above approach and treat angle as a base quantity
+with a base unit of radian and solid angle as its derived quantity.
+
+As those the above (at least for now) not a part of :term:`SI`, the plain angle and solid
+angle definitions can be found in the *units/generic* subdirectory. However, in the sake of correctness,
+they are consistently used in the dimensional recipes of all angle-based :term:`SI` quantities like
+torque or angular velocity.
+
+
+Radians and degrees support
+---------------------------
+
+Thanks to :ref:`framework/magnitudes:Magnitudes` usage the library provides efficient strong types for all
+angular types. This means that with the built-in support for magnitudes of :math:`\pi` we can provide
+accurate conversions between radians and degrees. The library also provides common trigonometric functions
+for angular quantities::
+
+    using namespace units::aliases::isq::si;
+
+    auto velocity = km_per_h<>(110);
+    auto rate_of_climb = m_per_s<>(-0.63657);
+    auto glide_ratio = velocity / -rate_of_climb;
+    auto glide_angle = asin(1 / glide_ratio);
+
+    std::cout << std::format("Glide ratio: {:%.1Q %q}\n", quantity_cast<units::one>(glide_ratio));
+    std::cout << "Glide angle:\n";
+    std::cout << std::format(" - {:%.4Q %q}\n", glide_angle);
+    std::cout << std::format(" - {:%.2Q %q}\n", quantity_cast<units::degree>(glide_angle));
+    std::cout << std::format(" - {:%.2Q %q}\n", quantity_cast<units::gradian>(glide_angle));
+
+The above program prints:
+
+.. code-block::
+
+    Glide ratio: 48.0 
+    Glide angle:
+     - 0.0208 rad
+     - 1.19 °
+     - 1.33 ᵍ
+
+.. note::
+
+    The above ``velocity`` and ``rate_of_climb`` in the production code should be modelled as
+    :ref:`framework/quantity_kinds:Quantity Kinds`.
+    A better but longer code can be found in the :ref:`examples/basics/glide_computer:glide_computer` example.