# Systems of Quantities¶

The physical units libraries on the market typically only scope on modeling one or more systems of units. However, this is not the only system kind to model. Another, and maybe even more important, system kind is a system of quantities.

Info

Please note that the mp-units is probably the first library on the Open Source market (in any programming language) that models the ISQ with all its definitions provided in ISO 80000. Please provide feedback if something looks odd or could be improved.

## Dimension is not enough to describe a quantity¶

Most of the products on the market are aware of physical dimensions. However, a dimension is not enough to describe a quantity. For example, let's see the following implementation:

class Box {
area base_;
length height_;
public:
Box(length l, length w, length h) : base_(l * w), height_(h) {}
// ...
};

Box my_box(2 * m, 3 * m, 1 * m);


How do you like such an interface? It turns out that in most existing strongly-typed libraries this is often the best we can do

Another typical question many users ask is how to deal with work and torque. Both of those have the same dimension but are different quantities.

A similar issue is related to figuring out what should be the result of:

auto res = 1 * Hz + 1 * Bq + 1 * Bd;


where:

• Hz (hertz) - unit of frequency
• Bq (becquerel) - unit of activity
• Bd (baud) - unit of modulation rate

All of those quantities have the same dimension, namely $$\mathsf{T}^{-1}$$, but probably it is not wise to allow adding, subtracting, or comparing them, as they describe vastly different physical properties.

If the above example seems too abstract, let's consider a fuel consumption (fuel volume divided by distance, e.g., 6.7 l/km) and an area. Again, both have the same dimension $$\mathsf{L}^{2}$$, but probably it wouldn't be wise to allow adding, subtracting, or comparing a fuel consumption of a car and the area of a football field. Such an operation does not have any physical sense and should fail to compile.

Important

More than one quantity may be defined for the same dimension:

• quantities of different kinds (e.g. frequency, modulation rate, activity, ...)
• quantities of the same kind (e.g. length, width, altitude, distance, radius, wavelength, position vector, ...)

It turns out that the above issues can't be solved correctly without proper modeling of a system of quantities.

## Quantities of the same kind¶

ISO 80000-1

• Quantities may be grouped together into categories of quantities that are mutually comparable
• Mutually comparable quantities are called quantities of the same kind
• Two or more quantities cannot be added or subtracted unless they belong to the same category of mutually comparable quantities
• Quantities of the same kind within a given system of quantities have the same quantity dimension
• Quantities of the same dimension are not necessarily of the same kind

The above quotes from ISO 80000 provide answers to all the issues above. Two quantities can't be added, subtracted, or compared unless they belong to the same kind. As frequency, activity, and modulation rate are different kinds, the expression provided above should not compile.

## System of quantities is not only about kinds¶

ISO 80000 specify hundreds of different quantities. There are plenty of different kinds provided and often each kind contains more than one quantity. In fact, it turns out that such quantities form a hierarchy of quantities of the same kind.

For example, here are all quantities of the kind length provided in the ISO 80000:

flowchart TD
length --- height[height, depth, altitude]
width --- thickness
width --- diameter
length --- path_length
path_length --- distance
length --- wavelength
length --- position_vector["position_vector<br>{vector}"]
length --- displacement["displacement<br>{vector}"]
radius --- radius_of_curvature

Each of the above quantities expresses some kind of length, and each can be measured with si::metre. However, each of them has different properties, usage, and sometimes even requires a different representation type (notice that position_vector and displacement are vector quantities).

Such a hierarchy helps us in defining arithmetics and conversion rules for various quantities of the same kind.

## Defining quantities¶

In the mp-units library all the information about the quantity is provided with the quantity_spec class template. In order to define a specific quantity a user should inherit a strong type from such an instantiation.

Tip

Quantity specification definitions benefit from an explicit object parameter added in C++23 to remove the need for CRTP idiom, which significantly simplifies the code. However, as C++23 is far from being mainstream today, a portability macro QUANTITY_SPEC() is provided and used consistently through the library to allow the code to compile with C++20 compilers, thanks to the CRTP usage under the hood.

See more in the C++ compiler support chapter.

For example, here is how the above quantity kind tree can be modeled in the library:

inline constexpr struct length final : quantity_spec<dim_length> {} length;
inline constexpr struct width final : quantity_spec<length> {} width;
inline constexpr auto breadth = width;
inline constexpr struct height final : quantity_spec<length> {} height;
inline constexpr auto depth = height;
inline constexpr auto altitude = height;
inline constexpr struct thickness final : quantity_spec<width> {} thickness;
inline constexpr struct diameter final : quantity_spec<width> {} diameter;
inline constexpr struct path_length final : quantity_spec<length> {} path_length;
inline constexpr auto arc_length = path_length;
inline constexpr struct distance final : quantity_spec<path_length> {} distance;
inline constexpr struct wavelength final : quantity_spec<length> {} wavelength;
inline constexpr struct position_vector final : quantity_spec<length, quantity_character::vector> {} position_vector;
inline constexpr struct displacement final : quantity_spec<length, quantity_character::vector> {} displacement;

inline constexpr struct length final : quantity_spec<length, dim_length> {} length;
inline constexpr struct width final : quantity_spec<width, length> {} width;
inline constexpr auto breadth = width;
inline constexpr struct height final : quantity_spec<height, length> {} height;
inline constexpr auto depth = height;
inline constexpr auto altitude = height;
inline constexpr struct thickness final : quantity_spec<thickness, width> {} thickness;
inline constexpr struct diameter final : quantity_spec<diameter, width> {} diameter;
inline constexpr struct path_length final : quantity_spec<path_length, length> {} path_length;
inline constexpr auto arc_length = path_length;
inline constexpr struct distance final : quantity_spec<distance, path_length> {} distance;
inline constexpr struct wavelength final : quantity_spec<wavelength, length> {} wavelength;
inline constexpr struct position_vector final : quantity_spec<position_vector, length, quantity_character::vector> {} position_vector;
inline constexpr struct displacement final : quantity_spec<displacement, length, quantity_character::vector> {} displacement;

QUANTITY_SPEC(length, dim_length);
QUANTITY_SPEC(width, length);
inline constexpr auto breadth = width;
QUANTITY_SPEC(height, length);
inline constexpr auto depth = height;
inline constexpr auto altitude = height;
QUANTITY_SPEC(thickness, width);
QUANTITY_SPEC(diameter, width);
QUANTITY_SPEC(path_length, length);
inline constexpr auto arc_length = path_length;
QUANTITY_SPEC(distance, path_length);
QUANTITY_SPEC(wavelength, length);
QUANTITY_SPEC(position_vector, length, quantity_character::vector);
QUANTITY_SPEC(displacement, length, quantity_character::vector);


Note

More information on how to define a system of quantities can be found in the "International System of Quantities (ISQ)" chapter.

## Comparing, adding, and subtracting quantities¶

ISO 80000 explicitly states that width and height are quantities of the same kind, and as such they:

• are mutually comparable,
• can be added and subtracted.

If we take the above for granted, the only reasonable result of 1 * width + 1 * height is 2 * length, where the result of length is known as a common quantity type. A result of such an equation is always the first common node in a hierarchy tree of the same kind. For example:

static_assert(common_quantity_spec(isq::width, isq::height) == isq::length);
static_assert(common_quantity_spec(isq::distance, isq::path_length) == isq::path_length);


## Converting between quantities¶

Based on the same hierarchy of quantities of kind length, we can define quantity conversion rules.

1. Implicit conversions

• every width is a length
• every radius is a width
static_assert(implicitly_convertible(isq::width, isq::length));

2. Explicit conversions

• not every length is a width
• not every width is a radius
static_assert(!implicitly_convertible(isq::length, isq::width));
static_assert(explicitly_convertible(isq::length, isq::width));

3. Explicit casts

• height is not a width
• both height and width are quantities of kind length
static_assert(!implicitly_convertible(isq::height, isq::width));
static_assert(!explicitly_convertible(isq::height, isq::width));
static_assert(castable(isq::height, isq::width));

4. No conversion

• time has nothing in common with length
static_assert(!implicitly_convertible(isq::time, isq::length));
static_assert(!explicitly_convertible(isq::time, isq::length));
static_assert(!castable(isq::time, isq::length));


## Hierarchies of derived quantities¶

Derived quantity equations often do not automatically form a hierarchy tree. This is why it is sometimes not obvious what such a tree should look like. Also, ISO explicitly states:

ISO/IEC Guide 99

The division of â€˜quantityâ€™ according to â€˜kind of quantityâ€™ is, to some extent, arbitrary.

The below presents some arbitrary hierarchy of derived quantities of kind energy:

flowchart TD
energy["energy<br>(mass * length<sup>2</sup> / time<sup>2</sup>)"]
energy --- mechanical_energy
mechanical_energy --- potential_energy
potential_energy --- gravitational_potential_energy["gravitational_potential_energy<br>(mass * acceleration_of_free_fall * height)"]
potential_energy --- elastic_potential_energy["elastic_potential_energy<br>(spring_constant * amount_of_compression<sup>2</sup>)"]
mechanical_energy --- kinetic_energy["kinetic_energy<br>(mass * speed<sup>2</sup>)"]
energy --- enthalpy
enthalpy --- internal_energy[internal_energy, thermodynamic_energy]
internal_energy --- Helmholtz_energy[Helmholtz_energy, Helmholtz_function]
enthalpy --- Gibbs_energy[Gibbs_energy, Gibbs_function]
energy --- active_energy

Notice, that even though all of those quantities have the same dimension and can be expressed in the same units, they have different quantity equations that can be used to create them implicitly:

• energy is the most generic one and thus can be created from base quantities of mass, length, and time. As those are also the roots of quantities of their kinds and all other quantities from their trees are implicitly convertible to them (we agreed on that "every width is a length" already), it means that an energy can be implicitly constructed from any quantity of mass, length, and time:

static_assert(implicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time), isq::energy));
static_assert(implicitly_convertible(isq::mass * pow<2>(isq::height) / pow<2>(isq::time), isq::energy));

• mechanical energy is a more "specialized" quantity than energy (not every energy is a mechanical energy). It is why an explicit cast is needed to convert from either energy or the results of its quantity equation:

static_assert(!implicitly_convertible(isq::energy, isq::mechanical_energy));
static_assert(explicitly_convertible(isq::energy, isq::mechanical_energy));
static_assert(!implicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time),
isq::mechanical_energy));
static_assert(explicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time),
isq::mechanical_energy));

• gravitational potential energy is not only even more specialized one but additionally, it is special in a way that it provides its own "constrained" quantity equation. Maybe not every mass * pow<2>(length) / pow<2>(time) is a gravitational potential energy, but every mass * acceleration_of_free_fall * height is.

static_assert(!implicitly_convertible(isq::energy, gravitational_potential_energy));
static_assert(explicitly_convertible(isq::energy, gravitational_potential_energy));
static_assert(!implicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time),
gravitational_potential_energy));
static_assert(explicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::time),
gravitational_potential_energy));
static_assert(implicitly_convertible(isq::mass * isq::acceleration_of_free_fall * isq::height,
gravitational_potential_energy));


## Modeling a quantity kind¶

In the physical units library, we also need an abstraction describing an entire family of quantities of the same kind. Such quantities have not only the same dimension but also can be expressed in the same units.

To annotate a quantity to represent its kind (and not just a hierarchy tree's root quantity) we introduced a kind_of<> specifier. For example, to express any quantity of length, we need to type kind_of<isq::length>.

Important

isq::length and kind_of<isq::length> are two different things.

Such an entity behaves as any quantity of its kind. This means that it is implicitly convertible to any quantity in a tree.

static_assert(!implicitly_convertible(isq::length, isq::height));
static_assert(implicitly_convertible(kind_of<isq::length>, isq::height));


Additionally, the result of operations on quantity kinds is also a quantity kind:

static_assert(same_type<kind_of<isq::length> / kind_of<isq::time>, kind_of<isq::length / isq::time>>);


However, if at least one equation's operand is not a quantity kind, the result becomes a "strong" quantity where all the kinds are converted to the hierarchy tree's root quantities:

static_assert(!same_type<kind_of<isq::length> / isq::time, kind_of<isq::length / isq::time>>);
static_assert(same_type<kind_of<isq::length> / isq::time, isq::length / isq::time>);


Info

Only a root quantity from the hierarchy tree or the one marked with is_kind specifier in the quantity_spec definition can be put as a template parameter to the kind_of specifier. For example, kind_of<isq::width> will fail to compile. However, we can call get_kind(q) to obtain a kind of any quantity:

static_assert(get_kind(isq::width) == kind_of<isq::length>);