Systems of Quantities¶

Most physical units libraries focus on modeling one or more systems of units. However an equally (or more) important abstraction is the system of quantities.

Info

mp-units is likely the first Open Source library (in any language) that models the ISQ with the full ISO 80000 definition set. Feedback is welcome.

Dimension is not enough to describe a quantity¶

Most libraries understand dimensions, yet a dimension alone does not fully describe a quantity. Consider:

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);

This interface is ambiguous. Many strongly typed libraries cannot do better

Another common question: how to differentiate work and torque? They share a dimension yet differ semantically.

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 have the same dimension \(\mathsf{T}^{-1}\), but adding or comparing them is meaningless.

Consider fuel consumption (fuel volume divided by distance, e.g. 6.7 l/km) vs an area. Both have dimension \(\mathsf{L}^{2}\) yet adding them is nonsensical and should fail.

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, ...)

These issues require 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

ISO 80000 answers the earlier questions: two quantities cannot be added, subtracted, or compared unless they are of the same kind. Thus frequency, activity, and modulation rate are incompatible.

System of quantities is not only about kinds¶

ISO 80000 specifies hundreds of quantities in many kinds; kinds often contain multiple quantities forming a hierarchy.

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

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

Each quantity above expresses some kind of length and can be measured with si::metre. Each has different semantics and sometimes a distinct representation (e.g. position_vector and displacement are vector quantities).

The hierarchy guides valid arithmetic and conversion rules for quantities of the same kind.

Defining quantities¶

All quantity information resides in quantity_spec. To define a quantity inherit a strong type from a suitable 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:

C++23C++20Portable

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 radius final : quantity_spec<width> {} radius;
inline constexpr struct radius_of_curvature final : quantity_spec<radius> {} radius_of_curvature;
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 radial_distance final : quantity_spec<distance> {} radial_distance;
inline constexpr struct wavelength final : quantity_spec<length> {} wavelength;
inline constexpr struct displacement final : quantity_spec<length, quantity_character::vector> {} displacement;
inline constexpr struct position_vector final : quantity_spec<displacement> {} position_vector;

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 radius final : quantity_spec<radius, width> {} radius;
inline constexpr struct radius_of_curvature final : quantity_spec<radius_of_curvature, radius> {} radius_of_curvature;
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 radial_distance final : quantity_spec<radial_distance, distance> {} radial_distance;
inline constexpr struct wavelength final : quantity_spec<wavelength, length> {} wavelength;
inline constexpr struct displacement final : quantity_spec<displacement, length, quantity_character::vector> {} displacement;
inline constexpr struct position_vector final : quantity_spec<position_vector, displacement> {} position_vector;

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(radius, width);
QUANTITY_SPEC(radius_of_curvature, radius);
QUANTITY_SPEC(path_length, length);
inline constexpr auto arc_length = path_length;
QUANTITY_SPEC(distance, path_length);
QUANTITY_SPEC(radial_distance, distance);
QUANTITY_SPEC(wavelength, length);
QUANTITY_SPEC(displacement, length, quantity_character::vector);
QUANTITY_SPEC(position_vector, displacement);

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 states that width and height are quantities of the same kind; therefore 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(get_common_quantity_spec(isq::width, isq::height) == isq::length);
static_assert(get_common_quantity_spec(isq::thickness, isq::radius) == isq::width);
static_assert(get_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.

Implicit conversions

every width is a length
every radius is a width

static_assert(implicitly_convertible(isq::width, isq::length));
static_assert(implicitly_convertible(isq::radius, isq::width));
static_assert(implicitly_convertible(isq::radius, isq::length));

Implicit conversions are allowed on copy-initialization:

void foo(quantity<isq::length[m]> q);

quantity<isq::width[m]> q1 = 42 * m;
quantity<isq::length[m]> q2 = q1;  // implicit quantity conversion
foo(q1);                           // implicit quantity conversion

Explicit conversions

not every length is a width
not every width is a radius

static_assert(!implicitly_convertible(isq::length, isq::width));
static_assert(!implicitly_convertible(isq::width, isq::radius));
static_assert(!implicitly_convertible(isq::length, isq::radius));
static_assert(explicitly_convertible(isq::length, isq::width));
static_assert(explicitly_convertible(isq::width, isq::radius));
static_assert(explicitly_convertible(isq::length, isq::radius));

Explicit conversions are forced by passing the quantity to a call operator of a quantity_spec type or by calling quantity's explicit constructor:

void foo(quantity<isq::height[m]> q);

quantity<isq::length[m]> q1 = 42 * m;
quantity<isq::height[m]> q2 = isq::height(q1);  // explicit quantity conversion
quantity<isq::height[m]> q3(q1);                // direct initialization
foo(isq::height(q1));                           // explicit quantity conversion

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));

Explicit casts are forced with a dedicated quantity_cast function:

void foo(quantity<isq::height[m]> q);

quantity<isq::width[m]> q1 = 42 * m;
quantity<isq::height[m]> q2 = quantity_cast<isq::height>(q1);  // explicit quantity cast
foo(quantity_cast<isq::height>(q1));                           // explicit quantity cast

No conversion

time has nothing in common with length

static_assert(!implicitly_convertible(isq::duration, isq::length));
static_assert(!explicitly_convertible(isq::duration, isq::length));
static_assert(!castable(isq::duration, isq::length));

Even the explicit casts will not force such a conversion:

void foo(quantity<isq::length[m]>);

quantity<isq::length[m]> q1 = 42 * s;    // Compile-time error
foo(quantity_cast<isq::length>(42 * s)); // Compile-time error

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 ‘quantity’ according to ‘kind of quantity’ is, to some extent, arbitrary.

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

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

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::duration), isq::energy));
static_assert(implicitly_convertible(isq::mass * pow<2>(isq::height) / pow<2>(isq::duration), 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::duration),
                                      isq::mechanical_energy));
static_assert(explicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::duration),
                                     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::duration),
                                      gravitational_potential_energy));
static_assert(explicitly_convertible(isq::mass * pow<2>(isq::length) / pow<2>(isq::duration),
                                     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::duration>, kind_of<isq::length / isq::duration>>);

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

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>);

Creating distinct quantity kinds with `is_kind`¶

While dimension-based type safety prevents many errors, sometimes quantities share the same dimension but represent fundamentally incompatible physical concepts. The is_kind specifier allows creating distinct quantity types that cannot be mixed even though they share the same dimension and quantity hierarchy tree.

When to use `is_kind`?¶

Use is_kind to create distinct subkinds within an existing quantity hierarchy when:

Multiple incompatible concepts need to share the same parent quantity's properties (unit or quantity type)
These concepts cannot be meaningfully added or compared to each other without explicit conversion
They represent different reference frames or measurement contexts, but derive from the same physical basis

The key insight: use is_kind when quantities need to inherit from a parent (quantity type, unit) but must be isolated from each other.

Common examples of subkinds within existing trees include:

Angular measure (rad), solid angular measure (sr), storage capacity (bit) — subkind of dimensionless
Fluid head and water head in hydraulic engineering — subkinds of height (dimension of length)

Defining a distinct kind¶

Important

The is_kind specifier creates subkinds within an existing quantity hierarchy tree, not independent trees. This allows the subkind to inherit properties from its parent:

Unit of measure: fluid head and water head inherit metre from height; angular measure inherits one from dimensionless
Quantity type: Subkinds inherit their parent's quantity type, which is crucial when they appear in derived quantities involving this quantity (e.g., sampling rate, tempo can use Hz because they properly model the dimensionless component divided by duration)

For quantities that should be completely independent (different dimension trees), define separate root quantities instead (e.g., frequency and activity are independent roots, not subkinds).

To create a distinct quantity kind as a subkind, add the is_kind specifier to the quantity_spec definition:

C++23C++20Portable

inline constexpr struct fluid_head final : quantity_spec<isq::height, is_kind> {} fluid_head;
inline constexpr struct water_head final : quantity_spec<isq::height, is_kind> {} water_head;
// Both inherit metre as unit and length as dimension from isq::height

inline constexpr struct fluid_head final : quantity_spec<fluid_head, isq::height, is_kind> {} fluid_head;
inline constexpr struct water_head final : quantity_spec<water_head, isq::height, is_kind> {} water_head;
// Both inherit metre as unit and length as dimension from isq::height

QUANTITY_SPEC(fluid_head, isq::height, is_kind);
QUANTITY_SPEC(water_head, isq::height, is_kind);
// Both inherit metre as unit and length as dimension from isq::height

Both fluid_head and water_head are subkinds of height (inheriting its dimension of length and unit of metre), but marking them with is_kind makes them distinct incompatible kinds that require explicit conversion.

Behavior of `is_kind` quantities¶

Quantities marked with is_kind behave differently from regular hierarchy members:

Cannot be implicitly converted to each other:

static_assert(!implicitly_convertible(fluid_head, water_head));
static_assert(!explicitly_convertible(fluid_head, water_head));
static_assert(!castable(fluid_head, water_head));

Cannot be added or compared directly:

quantity h_fluid = fluid_head(2 * m);
quantity h_water = water_head(10 * m);

// auto sum = h_fluid + h_water;  // Compile-time error!
// bool cmp = h_fluid < h_water;  // Compile-time error!

Require explicit conversion to base quantity:

To perform generic operations or conversions between kinds, explicit conversion to the base quantity is required:

// Convert to base quantity explicitly
quantity h1 = isq::height(h_fluid);  // explicit conversion required
quantity h2 = isq::height(h_water);  // explicit conversion required

// Now generic operations are possible
auto sum = h1 + h2;  // OK: both are isq::height

Warning

Implicit conversion from is_kind quantities to their base is not allowed:

quantity<isq::height[m]> h = h_fluid;  // Compile-time error!

Can be used with kind_of:

Unlike regular hierarchy members, is_kind quantities can be used with kind_of:

static_assert(get_kind(fluid_head) == kind_of<fluid_head>);
static_assert(get_kind(water_head) == kind_of<water_head>);
static_assert(get_kind(isq::height) == kind_of<isq::length>);
// static_assert(get_kind(isq::height) == kind_of<isq::height>);  // Compile-time error!

// Both are kinds of height, but different kinds
static_assert(get_kind(fluid_head) != get_kind(water_head));
static_assert(get_kind(fluid_head) != get_kind(isq::height));

Implementing physics-based conversions¶

When quantities are distinct kinds, domain-specific conversion functions should be provided to perform the correct physics-based transformations (if applicable):

// Define specific gravity as dimensionless
inline constexpr struct specific_gravity final : quantity_spec<dimensionless> {} specific_gravity;

// Physics: H_water = H_fluid * SG
constexpr QuantityOf<water_head> auto to_water_head(QuantityOf<fluid_head> auto h_fluid,
                                                    QuantityOf<specific_gravity> auto sg)
{
  return water_head(isq::height(h_fluid) * sg);
}

// Physics: H_fluid = H_water / SG
constexpr QuantityOf<fluid_head> auto to_fluid_head(QuantityOf<water_head> auto h_water,
                                                    QuantityOf<specific_gravity> auto sg)
{
  return fluid_head(isq::height(h_water) / sg);
}

This pattern:

Makes conversions explicit and visible in the code
Encodes the physics (specific gravity conversion formula)
Provides type-safe boundaries via QuantityOf constraints
Documents the relationship between different quantity kinds

Guidelines for using `is_kind`¶

Use is_kind when:

Quantities share a parent but have fundamentally different physical meanings
Adding or comparing them is physically nonsensical (e.g., plane angles + solid angles, fluid head + water head)
You need compile-time prevention of a known category of errors
Conversions between kinds either don't exist (plane vs solid angles) or require domain-specific formulas (fluid head ↔ water head via specific gravity)

Don't use is_kind when:

Quantities are naturally part of the same hierarchy (use regular quantity_spec hierarchy)
Conversions are just unit changes (use regular unit conversions)
The distinction is purely semantic without different physics (document in comments instead)

Tip

For a complete practical example demonstrating how is_kind prevents catastrophic engineering errors in hydraulic systems, see Tutorial 11: Preventing Confusion with Distinct Kinds.

Note

Special dimensionless quantity kinds like angular measure, solid angular measure, and storage capacity are discussed in detail in the Dimensionless Quantities chapter.

Systems of Quantities¶

Dimension is not enough to describe a quantity¶

Quantities of the same kind¶

System of quantities is not only about kinds¶

Defining quantities¶

Comparing, adding, and subtracting quantities¶

Converting between quantities¶

Hierarchies of derived quantities¶

Modeling a quantity kind¶

Creating distinct quantity kinds with is_kind¶

When to use is_kind?¶

Defining a distinct kind¶

Behavior of is_kind quantities¶

Implementing physics-based conversions¶

Guidelines for using is_kind¶

Creating distinct quantity kinds with `is_kind`¶

When to use `is_kind`?¶

Behavior of `is_kind` quantities¶

Guidelines for using `is_kind`¶