Units

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## Units

We define a unit as a set of base units each of which can be raised to an arbitrary rational exponent. Thus, the SI unit corresponding to the dimension of force is kg m s^-2, where kg, m, and s are base units. We use the notion of a unit system such as SI to specify the mapping from a dimension to a particular unit so that instead of specifying the base units explicitly, we can just ask for the representation of a dimension in a particular system.

Units are, like dimensions, purely compile-time variables with no associated value. Units obey the same algebra as dimensions do; the presence of the unit system serves to ensure that units having identical reduced dimension in different systems (like feet and meters) cannot be inadvertently mixed in computations.

There are two distinct types of systems that can be envisioned:

- Homogeneous systems : Systems which hold a linearly independent set of base units which can be used to represent many different dimensions. For example, the SI system has seven base dimensions and seven base units corresponding to them. It can represent any unit which uses only those seven base dimensions. Thus it is a homogeneous_system.
- Heterogeneous systems : Systems which store the exponents of every base unit involved are termed heterogeneous. Some units can only be represented in this way. For example, area in m ft is intrinsically heterogeneous, because the base units of meters and feet have identical dimensions. As a result, simply storing a dimension and a set of base units does not yield a unique solution. A practical example of the need for heterogeneous units, is an empirical equation used in aviation: H = (r/C)^2 where H is the radar beam height in feet and r is the radar range in nautical miles. In order to enforce dimensional correctness of this equation, the constant, C, must be expressed in nautical miles per foot^(1/2), mixing two distinct base units of length.

Units are implemented by the `unit` template class defined in `boost/units/unit.hpp` :

    template<class Dim,class System> class unit;

In addition to supporting the compile-time dimensional analysis operations, the +, -, *, and / runtime operators are provided for `unit` variables. Because the dimension associated with powers and roots must be computed at compile-time, it is not possible to provide overloads for `std::pow` that function correctly for `unit`s. These operations are supported through free functions `pow` and `root` that are templated on integer and `static_rational` values and can take as an argument any type for which the utility classes `power_typeof_helper` and `root_typeof_helper` have been defined.

### Base Units

Base units are defined much like base dimensions.

    template<class Derived, class Dimensions, long N> struct base_unit { ... };

Again negative ordinals are reserved.

As an example, in the following we will implement a subset of the SI unit system based on the fundamental dimensions given above, demonstrating all steps necessary for a completely functional system. First, we simply define a unit system that includes type definitions for commonly used units:

    struct meter_base_unit : base_unit<meter_base_unit, length_dimension, 1> { };
    struct kilogram_base_unit : base_unit<kilogram_base_unit, mass_dimension, 2> { };
    struct second_base_unit : base_unit<second_base_unit, time_dimension, 3> { };

    typedef make_system<
        meter_base_unit,
        kilogram_base_unit,
        second_base_unit>::type mks_system;

    /// unit typedefs
    typedef unit<dimensionless_type,mks_system>      dimensionless;

    typedef unit<length_dimension,mks_system>        length;
    typedef unit<mass_dimension,mks_system>          mass;
    typedef unit<time_dimension,mks_system>          time;

    typedef unit<area_dimension,mks_system>          area;
    typedef unit<energy_dimension,mks_system>        energy;

The macro `BOOST_UNITS_STATIC_CONSTANT` is provided in `boost/units/static_constant.hpp` to facilitate ODR- and thread-safe constant definition in header files. We then define some constants for the supported units to simplify variable definitions:

    /// unit constants
    BOOST_UNITS_STATIC_CONSTANT(meter,length);
    BOOST_UNITS_STATIC_CONSTANT(meters,length);
    BOOST_UNITS_STATIC_CONSTANT(kilogram,mass);
    BOOST_UNITS_STATIC_CONSTANT(kilograms,mass);
    BOOST_UNITS_STATIC_CONSTANT(second,time);
    BOOST_UNITS_STATIC_CONSTANT(seconds,time);

    BOOST_UNITS_STATIC_CONSTANT(square_meter,area);
    BOOST_UNITS_STATIC_CONSTANT(square_meters,area);
    BOOST_UNITS_STATIC_CONSTANT(joule,energy);
    BOOST_UNITS_STATIC_CONSTANT(joules,energy);

If support for textual output of units is desired, we can also specialize the `base_unit_info` class for each fundamental dimension tag:

    template<> struct base_unit_info<test::meter_base_unit>
    {
        static std::string name()               { return "meter"; }
        static std::string symbol()             { return "m"; }
    };

and similarly for `kilogram_base_unit` and `second_base_unit`. A future version of the library will provide a more flexible system allowing for internationalization through a facet/locale-type mechanism. The `name()` and `symbol()` methods of `base_unit_info` provide full and short names for the base unit. With these definitions, we have the rudimentary beginnings of our unit system, which can be used to determine reduced dimensions for arbitrary unit calculations.

### Scaled Base Units

Now, it is also possible to define a base unit as being a multiple of another base unit. For example, the way that `kilogram_base_unit` is actually defined by the library is along the following lines

    struct gram_base_unit : boost::units::base_unit<gram_base_unit, mass_dimension, 1> {};
    typedef scaled_base_unit<gram_base_unit, scale<10, static_rational<3> > > kilogram_base_unit;

This basically defines a kilogram as being 10^3 times a gram.

There are several advantages to this approach.

- It reflects the real meaning of these units better than treating them as independent units.
- If a conversion is defined between grams or kilograms and some other units, it will automatically work for both kilograms and grams, with only one specialization.
- Similarly, if the symbol for grams is defined as "g", then the symbol for kilograms will be "kg" without any extra effort.

### Scaled Units

We can also scale a `unit` as a whole, rather than scaling the individual base units which comprise it. For this purpose, we use the metafunction `make_scaled_unit`. The main motivation for this feature is the metric prefixes defined in `boost/units/systems/si/prefixes.hpp`.

A simple example of its usage would be.

    typedef make_scaled_unit<si::time, scale<10, static_rational<-9> > >::type nanosecond;

nanosecond is a specialization of `unit`, and can be used in a quantity normally.

    quantity<nanosecond> t(1.0 * si::seconds);
    std::cout << t << std::endl;    // prints 1e9 ns