boost_1_45_0/libs/units/doc/units.qbk

[library Boost.Units
  [quickbook 1.5]
  [version 1.1.0]
  [authors [Schabel, Matthias C.]]
  [authors [Watanabe, Steven]]
  [copyright 2003-2008 Matthias Christian Schabel, 2007-2010 Steven Watanabe]
  [license
      Distributed under the Boost Software License, Version 1.0.
      (See accompanying file LICENSE_1_0.txt or copy at
      [@http://www.boost.org/LICENSE_1_0.txt])
  ]
  [purpose zero-overhead compile-time dimensional analysis and unit computations]
]

[/ Some links to external sources.]
[def __boost [@http://www.boost.org/ Boost]]
[def __boostroot [@boost: Boost root]]
[def __boostlicense [@http://www.boost.org/LICENSE_1_0.txt Boost License]]
[def __boost_mpl [@http://www.boost.org/libs/mpl/doc/index.html Boost Metaprogramming Library]]

[def __mpl_forward_sequence [@http://www.boost.org/libs/mpl/doc/refmanual/forward-sequence.html MPL Forward Sequence]]

[/Links within this document.]
[def __ordinal [classref boost::units::ordinal ordinal]]
[def __dim [classref boost::units::dim dim]]
[def __static_rational [classref boost::units::static_rational static_rational]]
[def __make_dimension_list [classref boost::units::make_dimension_list make_dimension_list]]
[def __unit [classref boost::units::unit unit]]
[def __base_unit_info [classref boost::units::base_unit_info base_unit_info]]
[def __quantity [classref boost::units::quantity quantity]]
[def __conversion_helper [classref boost::units::conversion_helper conversion_helper]]
[def __absolute [classref boost::units::absolute absolute]]
[def __base_unit [classref boost::units::base_unit base_unit]]
[def __base_dimension [classref boost::units::base_dimension base_dimension]]
[def __scaled_base_unit [classref boost::units::scaled_base_unit base_unit]]
[def __make_scaled_unit [classref boost::units::make_scaled_unit make_scaled_unit]]

[def __unary_plus_typeof_helper [classref boost::units::unary_plus_typeof_helper unary_plus_typeof_helper]]
[def __unary_minus_typeof_helper [classref boost::units::unary_minus_typeof_helper unary_minus_typeof_helper]]
[def __add_typeof_helper [classref boost::units::add_typeof_helper add_typeof_helper]]
[def __subtract_typeof_helper [classref boost::units::subtract_typeof_helper subtract_typeof_helper]]
[def __multiply_typeof_helper [classref boost::units::multiply_typeof_helper multiply_typeof_helper]]
[def __divide_typeof_helper [classref boost::units::divide_typeof_helper divide_typeof_helper]]
[def __power_typeof_helper [classref boost::units::power_typeof_helper power_typeof_helper]]
[def __root_typeof_helper [classref boost::units::root_typeof_helper root_typeof_helper]]

[def __static_negate [classref boost::units::static_negate static_negate]]
[def __static_add [classref boost::units::static_add static_add]]
[def __static_subtract [classref boost::units::static_subtract static_subtract]]
[def __static_multiply [classref boost::units::static_multiply static_multiply]]
[def __static_divide [classref boost::units::static_divide static_divide]]
[def __static_power [classref boost::units::static_power static_power]]
[def __static_root [classref boost::units::static_root static_root]]

[def __get_dimension [classref boost::units::get_dimension get_dimension]]
[def __get_system [classref boost::units::get_system get_system]]

[def __pow [funcref boost::units::pow pow]]
[def __root [funcref boost::units::root root]]
[def __quantity_cast [funcref boost::units::quantity_cast quantity_cast]]

[def __from_value [memberref boost::units::quantity::from_value from_value]]
[def __value [memberref boost::units::quantity::value value]]

[def __reduce_unit [classref boost::units::reduce_unit reduce_unit]]
[def __unscale [classref boost::units::unscale unscale]]

[def __BOOST_UNITS_STATIC_CONSTANT [macroref BOOST_UNITS_STATIC_CONSTANT]]
[def __BOOST_UNITS_DEFINE_CONVERSION_FACTOR [macroref BOOST_UNITS_DEFINE_CONVERSION_FACTOR]]
[def __BOOST_UNITS_DEFINE_CONVERSION_FACTOR_TEMPLATE [macroref BOOST_UNITS_DEFINE_CONVERSION_FACTOR_TEMPLATE]]
[def __BOOST_UNITS_DEFAULT_CONVERSION [macroref BOOST_UNITS_DEFAULT_CONVERSION]]

[section:Introduction Introduction]

The Boost.Units library is a C++ implementation of dimensional analysis in a general 
and extensible manner, treating it as a generic compile-time metaprogramming problem. With appropriate
compiler optimization, no runtime execution cost is introduced, facilitating the use of this library to
provide dimension checking in performance-critical code. Support for units
and quantities (defined as a unit and associated value) for arbitrary unit system models and arbitrary  
value types is provided, as is a fine-grained general facility for unit conversions. Complete SI and CGS unit 
systems are provided, along with systems for 
angles measured in degrees, radians, gradians, and revolutions and 
systems for temperatures measured in Kelvin, degrees Celsius and degrees Fahrenheit. 
The library architecture has been designed with flexibility and extensibility in mind; demonstrations of the ease
of adding new units and unit conversions are provided in the examples.

In order to enable complex compile-time dimensional analysis calculations with no runtime overhead,
Boost.Units relies heavily on the [___boost_mpl] (MPL) and on template metaprogramming techniques, and is, as a consequence, 
fairly demanding of compiler compliance to ISO standards. At present, it has been successfully
compiled and tested on the following compilers/platforms :

# g++ 4.0.1 on Mac OSX 10.4
# Intel CC 9.1, 10.0, and 10.1 on Mac OSX 10.4
# g++ 3.4.4, 4.2.3, and 4.3.0 on Windows XP
# Microsoft Visual C++ 7.1, 8.0, and 9.0 on Windows XP
# Comeau 4.3.10.1 beta2 on Windows XP
# Metrowerks CodeWarrior 9.2 on Windows XP.  
# Sun CC 5.9 on Solaris and Linux

The following compilers/platforms are known *not* to work :

# g++ 3.3.x
# Microsoft Visual C++ 6.0 on Windows XP
# Microsoft Visual C++ 7.0 on Windows XP
# Metrowerks CodeWarrior 8.0 on Windows XP.
# All versions of Borland.

[endsect]

[section:Quick_Start Quick Start]

Before discussing the basics of the library, we first define a few terms that will be used frequently
in the following :

*  *Base dimension* : A base dimension is loosely defined as a measurable entity of interest; in conventional 
   dimensional analysis, base dimensions include length (\[L\]), mass (\[M\]), time (\[T\]), etc... but there is
   no specific restriction on what base dimensions can be used. Base dimensions are essentially a tag type and 
   provide no dimensional analysis functionality themselves.
*  *Dimension* : A collection of zero or more base dimensions, each
   potentially raised to a different rational power.
   For example, length = \[L\]^1, area = \[L\]^2, velocity = \[L\]^1/\[T\]^1, and
   energy = \[M\]^1 \[L\]^2/\[T\]^2 are all dimensions.
*  *Base unit* : A base unit represents a specific measure of a dimension. For example, while length is an abstract measure of
   distance, the meter is a concrete base unit of distance.  Conversions are defined using base units.
   Much like base dimensions, base units are a tag type used solely to define units and do not support dimensional
   analysis algebra.
*  *Unit* : A set of base units raised to rational exponents, e.g. m^1, kg^1, m^1/s^2.
*  *System* : A unit system is a collection of base units representing all the measurable entities of interest for a 
   specific problem. For example, the SI unit system defines seven base units : length (\[L\]) in meters, 
   mass (\[M\]) in kilograms, time (\[T\]) in seconds, current (\[I\]) in amperes, temperature (\[theta\]) in kelvin, 
   amount (\[N\]) in moles, and luminous intensity (\[J\]) in candelas. All measurable entities within the SI system can 
   be represented as products of various integer or rational powers of these seven base units.
*  *Quantity* : A quantity represents a concrete amount of a unit. Thus, while the meter is the base 
   unit of length in the SI system, 5.5 meters is a quantity of length in that system.

To begin, we present two short tutorials. [@../../libs/units/tutorial/tutorial_1.cpp Tutorial1] demonstrates the use of 
[@http://en.wikipedia.org/wiki/SI_units SI] units. After including the appropriate system headers 
and the headers for the various SI units we will need (all SI units can be included with  
[headerref boost/units/systems/si.hpp]) and for quantity I/O ([headerref boost/units/io.hpp]), we define 
a function that computes the work, in joules, done by exerting a force in newtons over a specified distance in meters 
and outputs the result to `std::cout`. The [___quantity] class accepts a second template parameter as its value type; 
this parameter defaults to
`double` if not otherwise specified. To demonstrate the ease of using user-defined types in dimensional
calculations, we also present code for computing the complex impedance using `std::complex<double>`
as the value type :

[import ../example/tutorial.cpp]

[tutorial_code]

The intent and function of the above code should be obvious; the output produced is :

[tutorial_output]

While this library attempts to make simple dimensional computations easy to code, it is in no way
tied to any particular unit system (SI or otherwise). Instead, it provides a highly flexible compile-time
system for dimensional analysis, supporting arbitrary collections of base dimensions, rational 
powers of units, and explicit quantity conversions. It accomplishes all of this via template metaprogramming techniques.
With modern optimizing compilers, this results in zero runtime overhead for quantity computations relative to the
same code without unit checking.

[endsect]

[section:Dimensional_Analysis Dimensional Analysis]

The concept of 
[@http://en.wikipedia.org/wiki/Dimensional_analysis dimensional analysis] 
is normally presented early on in introductory physics and engineering classes as a means of determining the  
correctness of an equation or computation by propagating the physical measurement
[@http://en.wikipedia.org/wiki/Units_of_measurement units]
of various quantities through the equation along with their numerical values. There are a number of standard 
unit systems in common use, the most prominent of which is the 
[@http://en.wikipedia.org/wiki/SI_units Systeme International] 
(also known as SI or MKS (meter-kilogram-second), which was a metric predecessor to the SI system named 
for three of the base units on which the system is based). The SI  
is the only official international standard unit system and is widely utilized in science and engineering. 
Other common systems include the [@http://en.wikipedia.org/wiki/Cgs_units CGS]
(centimeter-gram-second) system and the 
[@http://en.wikipedia.org/wiki/English_units English]
system still in use in some problem domains in the United States and elsewhere. In physics, 
there also exist a number of other systems that are in common use in specialized subdisciplines. These are 
collectively referred to as [@http://en.wikipedia.org/wiki/Natural_units natural units]. When 
quantities representing different measurables are combined, dimensional analysis provides the means of 
assessing the consistency of the resulting calculation. For example, the sum of two lengths is also a length, 
while the product of two lengths is an area, and the sum of a length and an area is undefined. The fact that the
arguments to many functions (such as exp, log, etc...) must be dimensionless quantities can be easily demonstrated by 
examining their series expansions in the context of dimensional analysis. This library facilitates the enforcement 
of this type of restriction in code involving dimensioned quantities where appropriate.

In the following discussion we view dimensional analysis as an abstraction in which an arbitrary set of 
[@http://en.wikipedia.org/wiki/Fundamental_units units] obey the rules of a specific algebra. 
We will refer to a pair of a base dimension and a rational exponent as a *fundamental dimension*, 
and a list composed of an arbitrary number of fundamental dimensions as a *composite dimension* or, simply,
*dimension*. In particular, given a set of [$../../libs/units/images/form_0.png] fundamental dimensions
denoted by [$../../libs/units/images/form_1.png] and a set of [$../../libs/units/images/form_0.png]
rational exponents [$../../libs/units/images/form_2.png], any possible (composite) dimension can be written
as [$../../libs/units/images/form_3.png].  

Composite dimensions obey the algebraic rules for dimensional analysis. In particular, for any scalar value,
[$../../libs/units/images/form_4.png], 
and composite dimensions [$../../libs/units/images/form_5.png]
and [$../../libs/units/images/form_6.png], where 
[$../../libs/units/images/form_7.png], we have:

[$../../libs/units/images/form_8.png]

Users of a dimensional analysis library should be able to specify an arbitrary list of base dimensions to 
produce a composite dimension. This potentially includes repeated tags. For example, 
it should be possible to express energy as [$../../libs/units/images/form_9.png], [$../../libs/units/images/form_10.png], 
[$../../libs/units/images/form_11.png], or any other permutation of mass, length, and time having aggregate exponents of 
1, 2, and -2, respectively.
In order to be able to perform computations on arbitrary sets of dimensions, 
all composite dimensions must be reducible to an unambiguous final composite dimension, which we will refer to as a 
*reduced dimension*, for which

# fundamental dimensions are consistently ordered
# dimensions with zero exponent are elided. Note that reduced dimensions never have more than 
  [$../../libs/units/images/form_0.png] base dimensions, one for each distinct fundamental dimension, but may have fewer.

In our implementation, base dimensions are associated with tag types. As we will ultimately 
represent composite dimensions as typelists, we must provide some mechanism for  sorting
base dimension tags in order to make it possible to convert an arbitrary composite dimension
into a reduced dimension.  For this purpose, we assign a unique integer to each base dimension.
The [___base_dimension] class (found in [headerref boost/units/base_dimension.hpp]) uses the
curiously recurring template pattern (CRTP) technique to ensure that ordinals specified for
base dimensions are unique:

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

With this, we can define the base dimensions for length, mass, and time as:

[import ../example/test_system.hpp]
[test_system_snippet_1]

It is important to note that the choice of order is completely arbitrary as long as each tag has a unique enumerable
value; non-unique ordinals are flagged as errors at compile-time. Negative ordinals are reserved for use by the library.  
To define composite dimensions corresponding to the base dimensions, we
simply create MPL-conformant typelists of fundamental dimensions by using the [___dim] class to encapsulate pairs of base dimensions 
and [___static_rational] exponents.  The [___make_dimension_list] class acts as a wrapper to ensure
that the resulting type is in the form of a reduced dimension:

[test_system_snippet_2]

This can also be easily accomplished using a convenience typedef provided by [___base_dimension]:

[test_system_snippet_3]

so that the above code is identical to the full typelist definition. Composite dimensions are similarly defined via a typelist:

[test_system_snippet_4]

A convenience class for composite dimensions with integer powers is also provided:

[test_system_snippet_5]

[endsect]

[section:Units 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 [headerref 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.

[section 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:

[test_system_snippet_6]

The macro [___BOOST_UNITS_STATIC_CONSTANT] is provided in [headerref 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:

[test_system_snippet_7]

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

[test_system_snippet_8]

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.

[endsect] [/section Base Units]

[section 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.

[endsect] [/section Scaled Base Units]

[section 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 [headerref 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

[endsect] [/section Scaled Units]

[endsect] [/section:Units Units]

[section:Quantities Quantities]

A *quantity* is defined as a value of an arbitrary value type that is associated with a specific unit. For example,
while meter is a unit, 3.0 meters is a quantity. Quantities obey two separate algebras: the native algebra for their 
value type, and the dimensional analysis algebra for the associated unit. In addition, algebraic operations are defined
between units and quantities to simplify the definition of quantities; it is effectively equivalent to algebra with
a unit-valued quantity.

Quantities are implemented by the [___quantity] template class defined in [headerref boost/units/quantity.hpp] :

    template<class Unit,class Y = double> class quantity;

This class is templated on both unit type (`Unit`) and value type (`Y`), with the latter defaulting to double-precision
floating point if not otherwise specified.  The value type must have a normal copy constructor and copy
assignment operator.  Operators +, -, *, and / are provided for algebraic operations between 
scalars and units, scalars and quantities, units and quantities, and between quantities. In addition, integral and
rational powers and roots can be computed using the [___pow]<R> and [___root]<R> functions. Finally,  
the standard set of boolean comparison operators ( `==, !=,  <, <=, >, and >=` ) are provided to allow 
comparison of quantities from the same unit system.  All operators simply delegate to the
corresponding operator of the value type if the units permit.

[section:Heterogeneous_Operators Heterogeneous Operators]

For most common value types, the result type of arithmetic operators is the same as the value type itself. For example, 
the sum of two double precision floating point numbers is another double precision floating point number. However, there 
are instances where this is not the case. A simple example is given by the [@http://en.wikipedia.org/wiki/Natural_number 
natural numbers] where the operator arithmetic obeys the following rules (using the standard notation for 
[@http://en.wikipedia.org/wiki/Number number systems]):

* [$../../libs/units/images/form_12.png]
* [$../../libs/units/images/form_13.png]
* [$../../libs/units/images/form_14.png]
* [$../../libs/units/images/form_15.png]

This library is designed to support arbitrary value type algebra for addition, subtraction, multiplication, division, and 
rational powers and roots.  It uses Boost.Typeof to deduce the result of these operators.  For compilers that
support `typeof`, the appropriate value type will be automatically deduced.  For compilers that do not provide
language support for `typeof` it is necessary to register all the types used.  For the case of natural numbers,
this would amount to something like the following:

    BOOST_TYPEOF_REGISTER_TYPE(natural);
    BOOST_TYPEOF_REGISTER_TYPE(integer);
    BOOST_TYPEOF_REGISTER_TYPE(rational);

[endsect]

[section:Conversions Conversions]

Conversion is only meaningful for quantities as it implies the presence of at 
least a multiplicative scale factor and, possibly, and affine linear offset.
Macros for simplifying the definition of conversions between units can be found in
[headerref boost/units/conversion.hpp] and [headerref boost/units/absolute.hpp]
(for affine conversions with offsets). 

The macro [___BOOST_UNITS_DEFINE_CONVERSION_FACTOR] specifies a scale
factor for conversion from the first unit type to the second.  The
first argument must be a [___base_unit].  The second argument
can be either a [___base_unit] or a [___unit].

Let's declare a simple base unit:

    struct foot_base_unit : base_unit<foot_base_unit, length_dimension, 10> { };

Now, we want to be able to convert feet to meters and vice versa.  The foot
is defined as exactly 0.3048 meters, so we can write the following

    BOOST_UNITS_DEFINE_CONVERSION_FACTOR(foot_base_unit, meter_base_unit, double, 0.3048);

Alternately, we could use the SI length `typedef`:

    BOOST_UNITS_DEFINE_CONVERSION_FACTOR(foot_base_unit, SI::length, double, 0.3048);

Since the SI unit of length is the meter, these two definitions are equivalent.
If these conversions have been defined, then converting between
scaled forms of these units will also automatically work.

The macro [___BOOST_UNITS_DEFAULT_CONVERSION] specifies a conversion
that will be applied to a base unit when no direct conversion is
possible.  This can be used to make arbitrary conversions work
with a single specialization:

    struct my_unit_tag : boost::units::base_unit<my_unit_tag, boost::units::force_type, 1> {};
    // define the conversion factor
    BOOST_UNITS_DEFINE_CONVERSION_FACTOR(my_unit_tag, SI::force, double, 3.14159265358979323846);
    // make conversion to SI the default.
    BOOST_UNITS_DEFAULT_CONVERSION(my_unit_tag, SI::force);

[endsect]

[section:Quantity_Construction_and_Conversion Construction and Conversion of Quantities]

This library is designed to emphasize safety above convenience when performing operations with dimensioned quantities.
Specifically, construction of quantities is required to fully specify both value and unit. Direct construction from a scalar value
is prohibited (though the static member function [___from_value] is provided to enable 
this functionality where it is necessary. In addition, a [___quantity_cast] to a reference allows direct access to the 
underlying value of a [___quantity] variable. An explicit constructor is provided to enable conversion between
dimensionally compatible quantities in different unit systems.  Implicit conversions between unit systems are
allowed only when the reduced units are identical, allowing, for example, trivial conversions between
equivalent units in different systems (such as SI seconds and CGS seconds) while simultaneously enabling
unintentional unit system mismatches to be caught at compile time and preventing potential loss of precision and
performance overhead from unintended conversions.  Assignment follows the same rules.
An exception is made for quantities for which the unit reduces to dimensionless; in this case, implicit conversion
to the underlying value type is allowed via class template specialization. Quantities of different value types are implicitly 
convertible only if the value types are themselves implicitly convertible. The [___quantity] class also defines
a `value()` member for directly accessing the underlying value.

To summarize, conversions are allowed under the following conditions :

* implicit conversion of `quantity<Unit,Y>` to `quantity<Unit,Z>` is allowed if `Y` and `Z` are implicitly convertible.
* assignment between `quantity<Unit,Y>` and `quantity<Unit,Z>` is allowed if `Y` and `Z` are implicitly convertible.
* explicit conversion between `quantity<Unit1,Y>` and `quantity<Unit2,Z>` is allowed if `Unit1` and `Unit2` have the same dimensions
    and if `Y` and `Z` are implicitly convertible.
* implicit conversion between `quantity<Unit1,Y>` and `quantity<Unit2,Z>` is allowed if `Unit1` 
    reduces to exactly the same combination of base units as `Unit2` and if `Y` and `Z` are convertible.
* assignment between `quantity<Unit1,Y>` and `quantity<Unit2,Z>` is allowed under the same
  conditions as implicit conversion.
* `quantity<Unit,Y>` can be directly constructed from a value of type `Y` using the static member function [___from_value]. Doing so, 
  naturally, bypasses any type-checking of the newly assigned value, so this method should be used only when absolutely necessary.

Of course, any time implicit conversion is allowed, an explicit conversion is
also legal.

Because dimensionless quantities have no associated units, they behave as normal scalars, and allow implicit conversion to and from 
the underlying value type or types that are convertible to/from that value type.

[endsect]

[endsect]

[section:Examples Examples]

[section:DimensionExample Dimension Example]

([@../../libs/units/example/dimension.cpp dimension.cpp])

By using MPL metafunctions and the template specializations for operations on composite dimensions
(defined in [headerref boost/units/dimension.hpp]) it is possible to perform compile time arithmetic
according to the dimensional analysis rules described [link boost_units.Dimensional_Analysis above]
to produce new composite dimensions :

[import ../example/dimension.cpp]

[dimension_snippet_1]

outputting (with symbol demangling, implemented in
[@boost:/boost/units/detail/utility.hpp utility.hpp])

[dimension_output]

[endsect]

[section:UnitExample Unit Example]

([@../../libs/units/example/unit.cpp unit.cpp])

This example demonstrates the use of the simple but functional unit system implemented in 
[@boost:/libs/units/example/test_system.hpp test_system.hpp]

[import ../example/unit.cpp]

[unit_snippet_1]

We can perform various algebraic operations on these units, resulting in the following output:

[unit_output]

[endsect]

[section:QuantityExample Quantity Example]

([@../../libs/units/example/quantity.cpp quantity.cpp])

This example demonstrates how to use quantities of our toy unit system :

[import ../example/quantity.cpp]

[quantity_snippet_1]

giving us the basic quantity functionality :

[quantity_output_double]

As a further demonstration of the flexibility of the system, we replace the `double` value type 
with a `std::complex<double>` value type (ignoring the question of the meaningfulness of
complex lengths and energies) :

[quantity_snippet_2]

and find that the code functions exactly as expected with no additional work, delegating operations 
to `std::complex<double>` and performing the appropriate dimensional analysis :

[quantity_output_complex]

[endsect]

[section:KitchenSinkExample Kitchen Sink Example using SI units]

([@../../libs/units/example/kitchen_sink.cpp kitchen_sink.cpp])

This example provides a fairly extensive set of tests covering most of the [___quantity] functionality.
It uses the SI unit system defined in [headerref boost/units/systems/si.hpp].

If we define a few units and associated quantities,

[import ../example/kitchen_sink.cpp]

[kitchen_sink_snippet_1]

the various algebraic operations between scalars, units, and quantities give 

[kitchen_sink_output_1]

Scalar/unit operations :

[kitchen_sink_output_2]

Unit/unit operations and integral/rational powers of units :

[kitchen_sink_output_3]

Scalar/quantity operations :

[kitchen_sink_output_4]

Unit/quantity operations :

[kitchen_sink_output_5]

Quantity/quantity operations and integral/rational powers of quantities :

[kitchen_sink_output_6]

Logical comparison operators are also defined between quantities :

[kitchen_sink_snippet_2]

giving

[kitchen_sink_output_7]

Implicit conversion is allowed between dimensionless quantities and their corresponding value types :

[kitchen_sink_snippet_3]

A generic function for computing mechanical work can be defined that takes force and distance arguments
in an arbitrary unit system and returns energy in the same system:

[kitchen_sink_function_snippet_3]

[kitchen_sink_snippet_4]

which functions as expected for SI quantities :

[kitchen_sink_output_9]

The ideal gas law can also be implemented in SI units :

[kitchen_sink_function_snippet_4]

[kitchen_sink_snippet_5]

with the resulting output :

[kitchen_sink_output_10]

Trigonometric and inverse trigonometric functions can be implemented for any unit system
that provides an angular base dimension. For radians, these functions are found in 
[headerref boost/units/cmath.hpp] These behave as one expects, with trigonometric functions
taking an angular quantity and returning a dimensionless quantity, while the inverse trigonometric functions
take a dimensionless quantity and return an angular quantity :

Defining a few angular quantities,

[kitchen_sink_snippet_6]

yields

[kitchen_sink_output_11]

Dealing with complex quantities is trivial. Here is the calculation of complex impedance :

[kitchen_sink_snippet_7]

giving

[kitchen_sink_output_12]

[section:UDT_Quantities User-defined value types]

User-defined value types that support the appropriate arithmetic operations are automatically supported
as quantity value types. The operators that are supported by default for quantity value types are unary plus, unary minus,
addition, subtraction, multiplication, division, equal-to, not-equal-to, less-than, less-or-equal-to, 
greater-than, and greater-or-equal-to. Support for rational powers and roots can be added by overloading
the [___power_typeof_helper] and [___root_typeof_helper] classes. Here we implement a user-defined `measurement`
class that models a numerical measurement with an associated measurement error and the appropriate algebra and
demonstrates its use as a quantity value type; the full code is found in [@../../libs/units/example/measurement.hpp measurement.hpp].

Then, defining some `measurement` [___quantity] variables

[kitchen_sink_snippet_8]

gives

[kitchen_sink_output_13]

If we implement the overloaded helper classes for rational powers and roots
then we can also compute rational powers of measurement quantities :

[kitchen_sink_output_14]

[endsect]

[endsect]

[section:ConversionExample Conversion Example]

([@../../libs/units/example/conversion.cpp conversion.cpp])

This example demonstrates the various allowed conversions between SI and CGS units. Defining some
quantities

[import ../example/conversion.cpp]

[conversion_snippet_1]

illustrates implicit conversion of quantities of different value types where implicit conversion 
of the value types themselves is allowed.  N.B. The conversion from double to int is treated
as an explicit conversion because there is no way to emulate the exact behavior of the built-in
conversion.  Explicit constructors allow conversions for two cases:

* explicit casting of a [___quantity] to a different `value_type` :

[conversion_snippet_3]

* and explicit casting of a [___quantity] to a different unit :

[conversion_snippet_4]

giving the following output :

[conversion_output_1]

A few more explicit unit system conversions :

[conversion_snippet_5]

which produces the following output:

[conversion_output_2]

[endsect]

[section:UDTExample User Defined Types]

([@../../libs/units/example/quaternion.cpp quaternion.cpp])

This example demonstrates the use of `boost::math::quaternion` as a value type for  [___quantity] and the converse.
For the first case, we first define specializations of [___power_typeof_helper] and [___root_typeof_helper] for 
powers and roots, respectively:

[import ../example/quaternion.cpp]

[quaternion_class_snippet_1a]

[quaternion_class_snippet_1b]

We can now declare a [___quantity] of a `quaternion` :

[quaternion_snippet_1]

so that all operations that are defined in the `quaternion` class behave correctly. If rational
powers were defined for this class, it would be possible to compute rational powers and roots with
no additional changes. 

[quaternion_output_1]

Now, if for some reason we preferred the [___quantity] to be the value type of the `quaternion` class we would have :

[quaternion_snippet_2]

Here, the unary plus and minus and addition and subtraction operators function correctly. Unfortunately, 
the multiplication and division operations fail because `quaternion` implements them in terms of the `*=` and 
`/=` operators, respectively, which are incapable of representing the heterogeneous unit algebra needed for 
quantities (an identical problem 
occurs with `std::complex<T>`, for the same reason). In order to compute rational powers and roots, we need to 
specialize [___power_typeof_helper] and [___root_typeof_helper] as follows:

[quaternion_class_snippet_2a]

[quaternion_class_snippet_2b]

giving:

[quaternion_output_2]

[endsect]

[section:ComplexExample Complex Example]

([@../../libs/units/example/complex.cpp complex.cpp])

This example demonstrates how to implement a replacement `complex` class that functions correctly both as a 
quantity value type and as a quantity container class, including heterogeneous multiplication and division 
operations and rational powers and roots. Naturally, heterogeneous operations are only supported on
compilers that implement `typeof`. The primary differences are that binary operations are not implemented
using the `op=` operators and use the utility classes [___add_typeof_helper], [___subtract_typeof_helper], 
[___multiply_typeof_helper], and [___divide_typeof_helper]. In addition, [___power_typeof_helper] and
[___root_typeof_helper] are defined for both cases :

[import ../example/complex.cpp]

[complex_class_snippet_1]

With this replacement `complex` class, we can declare a complex variable :

[complex_snippet_1]   

to get the correct behavior for all cases supported by [___quantity] with a `complex` value type :

[complex_output_1]

and, similarly, `complex` with a [___quantity] value type 

[complex_snippet_2]

gives

[complex_output_2]

[endsect]

[section:PerformanceExample Performance Example]

([@../../libs/units/example/performance.cpp performance.cpp])

This example provides an ad hoc performance test to verify that zero runtime overhead 
is incurred when using [___quantity] in place of `double`. Note that performance 
optimization and testing is not trivial, so some care must be taken in profiling. It 
is also critical to have a compiler capable of optimizing the many template instantiations
and inline calls effectively to achieve maximal performance. Zero overhead for this test
has been verified using gcc 4.0.1, and icc 9.0, 10.0, and 10.1 on Mac OS 10.4 and 10.5, and
using msvc 8.0 on Windows XP.

[endsect]

[section:RadarBeamHeightExample Radar Beam Height]

([@../../libs/units/example/radar_beam_height.cpp radar_beam_height.cpp])

[import ../example/radar_beam_height.cpp]

This example demonstrates the implementation of two non-SI units of length, the 
nautical mile :

[radar_beam_height_class_snippet_1]

and the imperial foot :

[radar_beam_height_class_snippet_2]

These units include conversions between themselves and the meter. Three functions
for computing radar beam height from radar range and the local earth radius are 
defined. The first takes arguments in one system and returns a value in the same 
system :

[radar_beam_height_function_snippet_1]

The second is similar, but is templated on return type, so that the arguments are
converted to the return unit system internally :

[radar_beam_height_function_snippet_2]

Finally, the third function is an empirical approximation that is only valid for
radar ranges specified in nautical miles, returning beam height in feet. This 
function uses the heterogeneous unit of nautical miles per square root of feet to
ensure dimensional correctness :

[radar_beam_height_function_snippet_3]

With these, we can compute radar beam height in various unit systems :

[radar_beam_height_snippet_1]

giving

[radar_beam_height_output]

[endsect]

[section:HeterogeneousUnitExample Heterogeneous Unit Example]

([@../../libs/units/example/heterogeneous_unit.cpp heterogeneous_unit.cpp])

[import ../example/heterogeneous_unit.cpp]

Mixed units and mixed unit conversions.

This code:

[heterogeneous_unit_snippet_1]

gives

[heterogeneous_unit_output_1]

Arbitrary conversions also work:

[heterogeneous_unit_snippet_2]

yielding

[heterogeneous_unit_output_2]

[endsect]

[section:AbsoluteRelativeTemperatureExample Absolute and Relative Temperature Example]

([@../../libs/units/example/temperature.cpp temperature.cpp])

[import ../example/temperature.cpp]

This example demonstrates using of absolute temperatures and relative temperature differences in Fahrenheit 
and converting between these and the Kelvin temperature scale. This issue touches on some surprisingly deep mathematical
concepts (see [@http://en.wikipedia.org/wiki/Affine_space Wikipedia] for a basic review), but for our purposes here, we
will simply observe that it is important to be able to differentiate between an absolute temperature measurement and a 
measurement of temperature difference. This is accomplished by using the [___absolute] wrapper class.

First we define a system using the predefined fahrenheit base unit:

[temperature_snippet_1]

Now we can create some quantities:

[temperature_snippet_3]

Note the use of [___absolute] to wrap a unit. The resulting output is:

[temperature_output_1]

[endsect]

[section:RuntimeConversionFactorExample Runtime Conversion Factor Example]

([@../../libs/units/example/runtime_conversion_factor.cpp runtime_conversion_factor.cpp])

[import ../example/runtime_conversion_factor.cpp]

The Boost.Units library does not require that the conversion factors be compile time constants,
as is demonstrated in this example:

[runtime_conversion_factor_snippet_1]

[endsect]

[section:UnitsWithNonbaseDimensions Units with Non-base Dimensions]

([@../../libs/units/example/non_base_dimension.cpp non_base_dimension.cpp])

[import ../example/non_base_dimension.cpp]

It is also possible to define base units that have derived rather than base dimensions:

[non_base_dimension_snippet_1]

[endsect]

[section:OutputForCompositeUnits Output for Composite Units]

([@../../libs/units/example/composite_output.cpp composite_output.cpp])

[import ../example/composite_output.cpp]

If a unit has a special name and/or symbol, the free functions `name_string` and
`symbol_string` can be overloaded directly. 

[composite_output_snippet_1]

In this case, any unit that reduces 
to the overloaded unit will be output with the replacement symbol. 

Special names and symbols for the SI and CGS unit systems are found in 
[headerref boost/units/systems/si/io.hpp] and [headerref boost/units/systems/cgs/io.hpp], 
respectively. If these headers are not included, the output will simply follow
default rules using the appropriate fundamental dimensions. 
Note that neither of these functions is defined for quantities 
because doing so would require making assumptions on how the corresponding value
type should be formatted.

Three `ostream` formatters, `symbol_format`, `name_format`, and `typename_format`
are provided for convenience. These select the textual representation of units 
provided by `symbol_string` or `name_string` in the first two cases, while the
latter returns a demangled typename for debugging purposes. Formatting of scaled
unit is also done correctly.

[endsect]

[section:autoscale Automatically Scaled Units]

It is often desirable to scale a [___unit] automatically, depending on its value,
to keep the integral part in a limited range, usually between 1 and 999.

For example, using [@http://en.wikipedia.org/wiki/Engineering_notation engineering notation prefixes],

 "1234.5 m" is more helpfully displayed as "1.234 km"
 "0.000000001234 m" is more clearly displayed as "1.2345 nanometer".

The iostream manipulators `engineering_prefixes` or `binary_prefixes` make this easy.

[import ../example/autoprefixes.cpp]

[autoprefixes_snippet_1]

(The complete set of [@http://physics.nist.gov/cuu/Units/prefixes.html engineering and scientific multiples]
is not used (not centi or deci for example), but only powers of ten that are multiples of three, 10^3).

Similarly, the equivalent [@http://en.wikipedia.org/wiki/Binary_prefixes binary prefixes]
used for displaying computing kilobytes, megabytes, gigabytes...

These are the 2^10 = 1024, 2^20 = 1 048 576, 2^30 ... multiples.

(See also [@http://physics.nist.gov/cuu/Units/binary.html Prefixes for binary multiples]

This scale is specified in IEC 60027-2, Second edition, 2000-11,
Letter symbols to be used in electrical technology -
Part 2: Telecommunications and electronics).

[autoprefixes_snippet_2]

But note that scalar dimensionless values, like int, float and double,
are *not* prefixed automatically by the engineering_prefix or binary_prefix iostream manipulators.

[autoprefixes_snippet_3]

You can output the name or symbol of a unit (rather than the most common quantity of a unit).

[autoprefixes_snippet_4]

Note too that all the formatting flags are persistent,
so that if you set engineering_prefix, then it applies to all future outputs,
until you select binary_prefix, or explicitly switch autoprefix off.
You can specify no prefix (the default of course) in two ways:

[autoprefixes_snippet_5]

And you can get the format flags for diagnosing problems.

[autoprefixes_snippet_6]

[endsect] [/section:autoscale Automatically Scaled Units]

[section:ConversionFactor Conversion Factor]

This code demonstrates the use of the `conversion_factor` free function to determine
the scale factor between two units.

([@../../libs/units/example/conversion_factor.cpp conversion_factor.cpp])

[import ../example/conversion_factor.cpp]

[conversion_factor_snippet_1]

Produces

[conversion_factor_output]

[endsect]

[section:RuntimeUnits Runtime Units]

([@../../libs/units/example/runtime_unit.cpp runtime_unit.cpp])

[import ../example/runtime_unit.cpp]

This example shows how to implement an interface that
allow different units at runtime while still maintaining
type safety for internal calculations.

[runtime_unit_snippet_1]

[endsect]

[section:lambda Interoperability with Boost.Lambda]

([@../../libs/units/example/lambda.cpp lambda.cpp])

[import ../example/lambda.cpp]

The header [headerref boost/units/lambda.hpp] provides overloads
and specializations needed to make Boost.Units usable with the
Boost.Lambda library.

[lambda_snippet_1]

[endsect]

[endsect]

[section:Utilities Utilities]

Relatively complete SI and CGS unit systems are provided in [headerref boost/units/systems/si.hpp] and
[headerref boost/units/systems/cgs.hpp], respectively. 

[section:Metaprogramming_Classes Metaprogramming Classes]

    template<long N> struct ordinal<N>;

    template<typename T,typename V> struct get_tag< dim<T,V> >;
    template<typename T,typename V> struct get_value< dim<T,V> >;
    template<class S,class DT> struct get_system_tag_of_dim<S,DT>;
    template<typename Seq> struct make_dimension_list<Seq>;
    template<class DT> struct fundamental_dimension<DT>;
    template<class DT1,int E1,...> struct composite_dimension<DT1,E1,...>;

    template<class Dim,class System> struct get_dimension< unit<Dim,System> >;
    template<class Unit,class Y> struct get_dimension< quantity<Unit,Y> >;
    template<class Dim,class System> struct get_system< unit<Dim,System> >;
    template<class Unit,class Y> struct get_system quantity<Unit,Y> >;

    struct dimensionless_type;
    template<class System> struct dimensionless_unit<System>;
    template<class System,class Y> struct dimensionless_quantity<System,Y>;

    struct implicitly_convertible;
    struct trivial_conversion;
    template<class T,class S1,class S2> struct base_unit_converter<T,S1,S2>;

    template<class Q1,class Q2> class conversion_helper<Q1,Q2>;

[endsect]

[section:Metaprogramming_Predicates Metaprogramming Predicates]

    template<typename T,typename V> struct is_dim< dim<T,V> >;
    template<typename T,typename V> struct is_empty_dim< dim<T,V> >;

    template<typename Seq> struct is_dimension_list<Seq>;

    template<class S> struct is_system< homogeneous_system<S> >;
    template<class S> struct is_system< heterogeneous_system<S> >;
    template<class S> struct is_homogeneous_system< homogeneous_system<S> >;
    template<class S> struct is_heterogeneous_system< heterogeneous_system<S> >;

    template<class Dim,class System> struct is_unit< unit<Dim,System> >;
    template<class Dim,class System> struct is_unit_of_system< unit<Dim,System>,System >;
    template<class Dim,class System> struct is_unit_of_dimension< unit<Dim,System>,Dim >;

    template<class Unit,class Y> struct is_quantity< quantity<Unit,Y> >;
    template<class Dim,class System,class Y> struct is_quantity_of_system< quantity<unit<Dim,System>,Y>,System >;
    template<class Dim,class System,class Y> struct is_quantity_of_dimension< quantity<unit<Dim,System>,Y>,Dim >;

    template<class System> struct is_dimensionless< unit<dimensionless_type,System> >;
    template<class System> struct is_dimensionless_unit< unit<dimensionless_type,System> >;
    template<class System,class Y> struct is_dimensionless< quantity<unit<dimensionless_type,System>,Y> >;
    template<class System,class Y> struct is_dimensionless_quantity< quantity<unit<dimensionless_type,System>,Y> >; 

[endsect]

[endsect]

[section:Reference Reference]

[xinclude units_reference.xml]

[xinclude dimensions_reference.xml]
[xinclude si_reference.xml]
[xinclude cgs_reference.xml]
[xinclude trig_reference.xml]
[xinclude temperature_reference.xml]
[xinclude abstract_reference.xml]

[section Base Units by Category]

[xinclude angle_base_units_reference.xml]
[xinclude astronomical_base_units_reference.xml]
[xinclude cgs_base_units_reference.xml]
[xinclude imperial_base_units_reference.xml]
[xinclude metric_base_units_reference.xml]
[xinclude si_base_units_reference.xml]
[xinclude temperature_base_units_reference.xml]
[xinclude us_base_units_reference.xml]

[endsect]

[section Alphabetical Listing of Base Units]
[include base_units.qbk]
[endsect]

[endsect]

[section:Installation Installation]

The core header files are located in `boost/units`. Unit system headers are 
located in `<boost/units/systems>`. There are no source files for the library
itself - the library is header-only. Example programs demonstrating various aspects of the library can be found in 
`boost/libs/units/example`. Programs for unit testing are provided in `boost/libs/units/test`.

[endsect]

[section:FAQ FAQ]

[section:Distinguishing_Quantities_With_Same_Units
How does one distinguish between quantities that are physically different but have the same units (such as 
energy and torque)?]

Because Boost.Units includes plane and solid angle units in the SI system, torque and energy 
are, in fact, distinguishable (see [@http://en.wikipedia.org/wiki/SI_units torque]). 
In addition, energy is a true
[@http://mathworld.wolfram.com/Scalar.html scalar] quantity, while torque, despite
having the same units as energy if plane angle is not included, is in fact a 
[@http://mathworld.wolfram.com/Pseudovector.html pseudovector]. Thus, a value type representing pseudovectors 
and encapsulating their algebra could also be implemented. 

There are,
however, a few SI units that are dimensionally indistinguishable within the SI system. These
include the [@http://en.wikipedia.org/wiki/Becquerel becquerel], which has units identical to
frequency (Hz), and the [@http://en.wikipedia.org/wiki/Sievert sievert], which is degenerate
with the [@http://en.wikipedia.org/wiki/Gray_%28unit%29 gray]. In cases such as this, 
the proper way to treat this difference is to recognize that expanding the set of base dimensions
can provide disambiguation.  For example, adding a base dimension for radioactive decays would 
allow the becquerel to be written as decays/second, differentiating it from the signature of hertz,
which is simply 1/second.

[endsect]

[section:Angle_Are_Units Angles are treated as units]

If you don't like this, you can just ignore the angle units and 
go on your merry way (periodically screwing up when a routine wants degrees and you give it 
radians instead...)

[endsect]

[section:Why_Homogeneous_Systems Why are there homogeneous systems?  Aren't heterogeneous systems sufficient?]

Consider the following code:

    cout << sin(asin(180.0 * degrees));

What should this print?  If only heterogeneous
systems are available it would print 3.14159+ rad
Why?  Well, `asin` would return a `quantity<dimensionless>`
effectively losing the information that degrees
are being used.  In order to propogate this extra information
we need homogeneous systems.

[endsect]

[section:NoConstructorFromValueType Why can't I construct a quantity directly from the value type?]

This only breaks generic code--which ought to break anyway.  The only
literal value that ought to be converted to a quantity by generic code
is zero, which should be handled by the default constructor. In addition,
consider the search and replace problem allowing this poses:

	quantity<si::length>    q(1.0);
	
Here, the intent is clear - we want a length of one in the SI system, which is one meter. However,
imagine some well-intentioned coder attempting to reuse this code, but to have it perform the
calculations in the CGS unit system instead. After searching for `si::` and replacing it with `cgs::` ,
we have:

	quantity<cgs::length>	q(1.0);
	
Unfortunately, the meaning of this statement has suddenly changed from one meter to one centimeter. In
contrast, as implemented, we begin with:

	quantity<si::length>	q(1.0*si::meter);
	
and, after search and replace:

	quantity<cgs::length>	q(1.0*cgs::meter);
	
which gives us an error. Even if the code has a @using namespace boost::units::si; declaration, the latter
is still safe, with:

	using namespace boost::units::si;
	quantity<length>	q(1.0*meter);
	
going to

	using namespace boost::units::cgs;
	quantity<length>	q(1.0*meter);
	
The latter will involve an explicit conversion from meters to centimeters, but the value remains correct.

[endsect]

[section:ExplicitConversions Why are conversions explicit by default?]

Safety and the potential for unintended conversions leading to precision loss and hidden performance costs.
Options are provided for forcing implicit conversions between specific units to be allowed.

[endsect]

[endsect]

[section:Acknowledgements Acknowledgements]

Matthias C. Schabel would like to acknowledge the Department of Defense for its support of this work under 
the Prostate Cancer Research Program New Investigator Award W81XWH-04-1-0042 and the National Institutes of Health for their
support of this work under the NIBIB Mentored Quantitative Research Development Award K25EB005077.

Thanks to David Walthall for his assistance in debugging and testing on a variety of platforms and Torsten Maehne for
his work on interfacing the Boost Units and Boost Lambda libraries.

Thanks to:

* Paul Bristow, 
* Michael Fawcett, 
* Ben FrantzDale, 
* Ron Garcia,
* David Greene,
* Peder Holt,
* Janek Kozicki, 
* Andy Little,
* Kevin Lynch,
* Torsten Maehne
* Noah Roberts,
* Andrey Semashev,
* David Walthall,
* Deane Yang, 

and all the members of the Boost mailing list who provided their input into 
the design and implementation of this library.

[endsect] [/section:Acknowledgements Acknowledgements]

[section:HelpWanted Help Wanted]

 Any help in the following areas would be much appreciated:

* testing on other compilers and operating systems
* performance testing on various architectures
* tutorials 

[endsect]

[section:version_id Version Info]

__boostroot

Last edit to Quickbook file __FILENAME__ was at __TIME__ on __DATE__.

[tip This should appear on the pdf version (but may be redundant on html).]
[/ Useful on pdf version. See also Last revised timestamp on first page of html version.]
[/See also Adobe Reader pdf File Properties for creation date, and PDF producer, version and page count.]

[endsect] [/section:version_id Version Info]

[section:ReleaseNotes Release Notes]

1.2 (March 2010)

* Added autoprefix ready for Boost 1.43

1.0.0 (August 1, 2008) :

* Initial release with Boost 1.36

0.7.1 (March 14, 2007) :

* Boost.Typeof emulation support.
* attempting to rebind a heterogeneous_system to a different set of dimensions now fails.
* cmath.hpp now works with como-win32.
* minor changes to the tests and examples to make msvc 7.1 happy.

0.7.0 (March 13, 2007) :

* heterogeneous and mixed system functionality added.
* added fine-grained implicit unit conversion on a per fundamental dimension basis.
* added a number of utility metafunction classes and predicates.
* [headerref boost/units/operators.hpp] now uses `BOOST_TYPEOF` when possible.
* angular units added in [headerref boost/units/systems/angle/gradians.hpp]
  and [headerref boost/units/systems/angle/gradians.hpp].
  Implicit conversion of radians between trigonometric, SI, and CGS systems is allowed.
* a variety of [___unit] and [___quantity] tests added.
* examples now provide self-tests.

0.6.2 (February 22, 2007) :

*  changed template order in `unit` so dimension precedes unit system
*  added `homogeneous_system<S>` for unit systems
*  incorporated changes to [headerref boost/units/dimension.hpp] (compile-time sorting by predicate), 
   [headerref boost/units/conversion.hpp] (thread-safe implementation of quantity conversions), 
   and [headerref boost/units/io.hpp] (now works with any `std::basic_ostream`) by SW
* added abstract units in [headerref boost/units/systems/abstract.hpp] to allow abstract dimensional
  analysis
* new example demonstrating implementation of code based on requirements from 
  Michael Fawcett ([@../../libs/units/example/radar_beam_height.cpp radar_beam_height.cpp])

0.6.1 (February 13, 2007) :

* added metafunctions to test if a type is 
    * a valid dimension list (`is_dimension_list<D>`)
    * a unit (`is_unit<T>` and `is_unit_of_system<U,System>`)
    * a quantity (`is_quantity<T>` and `is_quantity_of_system<Q,System>`) 
* quantity conversion factor is now computed at compile time 
* static constants now avoid ODR problems
* unit_example_14.cpp now uses Boost.Timer
* numerous minor fixes suggested by SW

0.6.0 (February 8, 2007) :

* incorporated Steven Watanabe's optimized code for dimension.hpp, leading to *dramatic*
  decreases in compilation time (nearly a factor of 10 for unit_example_4.cpp in my tests).

0.5.8 (February 7, 2007) :

* fixed `#include` in [headerref boost/units/systems/si/base.hpp] (thanks to Michael Fawcett and 
  Steven Watanabe)
* removed references to obsolete `base_type` in [___unit_info] (thanks to Michael Fawcett)
* moved functions in [headerref boost/units/cmath.hpp] into `boost::units` namespace 
  (thanks to Steven Watanabe)
* fixed `#include` guards to be consistently named `BOOST_UNITS_XXX` (thanks to Steven 
  Watanabe)

0.5.7 (February 5, 2007) :

* changed quantity conversion helper to increase flexibility
* minor documentation changes
* submitted for formal review as a Boost library

0.5.6 (January 22, 2007) :

* added IEEE 1541 standard binary prefixes along with SI prefixes to and extended algebra of
  `scale` and `scaled_value` classes (thanks to Kevin Lynch)
* split SI units into separate header files to minimize the "kitchen sink" include problem
  (thanks to Janek Kozicki)
* added convenience classes for declaring fundamental dimensions and composite dimensions 
   with integral powers (`fundamental_dimension` and `composite_dimension` respectively)

0.5.5 (January 18, 2007) :

* template parameter order in `quantity` switched and default `value_type` of `double` added
  (thanks to Andrey Semashev and Paul Bristow)
* added implicit `value_type` conversion where allowed (thanks to Andrey Semashev)
* added `quantity_cast` for three cases (thanks to Andrey Semashev):
    * constructing `quantity` from raw `value_type`
    * casting from one `value_type` to another
    * casting from one `unit` to another (where conversion is allowed) 
* added` metre` and `metres` and related constants to the SI system for the convenience of
  our Commonwealth friends...

0.5.4 (January 12, 2007) :

* completely reimplemented unit conversion to allow for arbitrary unit conversions
  between systems
* strict quantity construction is default; quantities can be constructed from bare values 
  by using static member `from_value`

0.5.3 (December 12, 2006) :

* added Boost.Serialization support to `unit` and `quantity` classes
* added option to enforce strict construction of quantities (only constructible
  by multiplication of scalar by unit or quantity by unit) by preprocessor
  `MCS_STRICT_QUANTITY_CONSTRUCTION` switch

0.5.2 (December 4, 2006) :

* added `<cmath>` wrappers in the `std` namespace for functions that can support quantities 

0.5.1 (November 3, 2006) :

* converted to Boost Software License
* boostified directory structure and file paths

0.5 (November 2, 2006) :

* completely reimplemented SI and CGS unit systems and changed syntax for quantities
* significantly streamlined `pow` and `root` so for most applications it is only
  necessary to define `power_typeof_helper` and `root_typeof_helper` to gain this
  functionality
* added a selection of physical constants from the CODATA tables
* added a skeleton `complex` class that correctly supports both `complex<quantity<Y,Unit> >`
  and `quantity<complex<Y>,Unit>` as an example
* investigate using Boost.Typeof for compilers that do not support `typeof`

0.4 (October 13, 2006) : 

* `pow<R>` and `root<R>` improved for user-defined types
* added unary + and unary - operators
* added new example of interfacing with `boost::math::quaternion`
* added optional preprocessor switch to enable implicit unit conversions
  (`BOOST_UNITS_ENABLE_IMPLICIT_UNIT_CONVERSIONS`) 

0.3 (September 6, 2006) :

* Support for `op(X x,Y y)` for g++ added. This is automatically
  active when compiling with gcc and can be optionally enabled by defining the preprocessor
  constant `BOOST_UNITS_HAS_TYPEOF`

0.2 (September 4, 2006) : Second alpha release based on slightly modified code from 0.1 release

0.1 (December 13, 2003)  : written as a Boost demonstration of MPL-based dimensional analysis
in 2003.

[endsect]

[section:TODO TODO]

* Document concepts
* Implementation of I/O is rudimentary; consider methods of i18n using facets
* Consider runtime variant, perhaps using overload like `quantity<runtime,Y>`

[endsect] [/section:TODO TODO]