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| <div class="titlepage"><div><div><h2 class="title" style="clear: both"> |
| <a name="boost_units.Dimensional_Analysis"></a><a class="link" href="Dimensional_Analysis.html" title="Dimensional Analysis">Dimensional Analysis</a> |
| </h2></div></div></div> |
| <p> |
| The concept of <a href="http://en.wikipedia.org/wiki/Dimensional_analysis" target="_top">dimensional |
| analysis</a> 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 <a href="http://en.wikipedia.org/wiki/Units_of_measurement" target="_top">units</a> |
| 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 <a href="http://en.wikipedia.org/wiki/SI_units" target="_top">Systeme |
| International</a> (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 <a href="http://en.wikipedia.org/wiki/Cgs_units" target="_top">CGS</a> |
| (centimeter-gram-second) system and the <a href="http://en.wikipedia.org/wiki/English_units" target="_top">English</a> |
| 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 <a href="http://en.wikipedia.org/wiki/Natural_units" target="_top">natural units</a>. 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. |
| </p> |
| <p> |
| In the following discussion we view dimensional analysis as an abstraction |
| in which an arbitrary set of <a href="http://en.wikipedia.org/wiki/Fundamental_units" target="_top">units</a> |
| obey the rules of a specific algebra. We will refer to a pair of a base dimension |
| and a rational exponent as a <span class="bold"><strong>fundamental dimension</strong></span>, |
| and a list composed of an arbitrary number of fundamental dimensions as a |
| <span class="bold"><strong>composite dimension</strong></span> or, simply, <span class="bold"><strong>dimension</strong></span>. In particular, given a set of <span class="inlinemediaobject"><img src="../../../libs/units/images/form_0.png" alt="form_0"></span> fundamental dimensions denoted by <span class="inlinemediaobject"><img src="../../../libs/units/images/form_1.png" alt="form_1"></span> and a set of <span class="inlinemediaobject"><img src="../../../libs/units/images/form_0.png" alt="form_0"></span> rational exponents <span class="inlinemediaobject"><img src="../../../libs/units/images/form_2.png" alt="form_2"></span>, any possible (composite) dimension can be written as |
| <span class="inlinemediaobject"><img src="../../../libs/units/images/form_3.png" alt="form_3"></span>. |
| </p> |
| <p> |
| Composite dimensions obey the algebraic rules for dimensional analysis. In |
| particular, for any scalar value, <span class="inlinemediaobject"><img src="../../../libs/units/images/form_4.png" alt="form_4"></span>, and composite dimensions <span class="inlinemediaobject"><img src="../../../libs/units/images/form_5.png" alt="form_5"></span> and <span class="inlinemediaobject"><img src="../../../libs/units/images/form_6.png" alt="form_6"></span>, where <span class="inlinemediaobject"><img src="../../../libs/units/images/form_7.png" alt="form_7"></span>, we have: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../libs/units/images/form_8.png" alt="form_8"></span> |
| </p> |
| <p> |
| 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 <span class="inlinemediaobject"><img src="../../../libs/units/images/form_9.png" alt="form_9"></span>, <span class="inlinemediaobject"><img src="../../../libs/units/images/form_10.png" alt="form_10"></span>, <span class="inlinemediaobject"><img src="../../../libs/units/images/form_11.png" alt="form_11"></span>, 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 <span class="bold"><strong>reduced dimension</strong></span>, for which |
| </p> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| fundamental dimensions are consistently ordered |
| </li> |
| <li class="listitem"> |
| dimensions with zero exponent are elided. Note that reduced dimensions |
| never have more than <span class="inlinemediaobject"><img src="../../../libs/units/images/form_0.png" alt="form_0"></span> base dimensions, one for each distinct fundamental |
| dimension, but may have fewer. |
| </li> |
| </ol></div> |
| <p> |
| 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 <span class="underline"><code class="computeroutput"><a class="link" href="../boost/units/base_dimension.html" title="Class template base_dimension">base_dimension</a></code></span> class |
| (found in <code class="computeroutput"><a class="link" href="Reference.html#header.boost.units.base_dimension_hpp" title="Header <boost/units/base_dimension.hpp>">boost/units/base_dimension.hpp</a></code>) |
| uses the curiously recurring template pattern (CRTP) technique to ensure that |
| ordinals specified for base dimensions are unique: |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Derived</span><span class="special">,</span> <span class="keyword">long</span> <span class="identifier">N</span><span class="special">></span> <span class="keyword">struct</span> <span class="identifier">base_dimension</span> <span class="special">{</span> <span class="special">...</span> <span class="special">};</span> |
| </pre> |
| <p> |
| With this, we can define the base dimensions for length, mass, and time as: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="comment">/// base dimension of length |
| </span><span class="keyword">struct</span> <span class="identifier">length_base_dimension</span> <span class="special">:</span> <span class="identifier">base_dimension</span><span class="special"><</span><span class="identifier">length_base_dimension</span><span class="special">,</span><span class="number">1</span><span class="special">></span> <span class="special">{</span> <span class="special">};</span> |
| <span class="comment">/// base dimension of mass |
| </span><span class="keyword">struct</span> <span class="identifier">mass_base_dimension</span> <span class="special">:</span> <span class="identifier">base_dimension</span><span class="special"><</span><span class="identifier">mass_base_dimension</span><span class="special">,</span><span class="number">2</span><span class="special">></span> <span class="special">{</span> <span class="special">};</span> |
| <span class="comment">/// base dimension of time |
| </span><span class="keyword">struct</span> <span class="identifier">time_base_dimension</span> <span class="special">:</span> <span class="identifier">base_dimension</span><span class="special"><</span><span class="identifier">time_base_dimension</span><span class="special">,</span><span class="number">3</span><span class="special">></span> <span class="special">{</span> <span class="special">};</span> |
| |
| </pre> |
| <p> |
| </p> |
| <p> |
| 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 <span class="underline"><code class="computeroutput"><a class="link" href="../boost/units/dim.html" title="Struct template dim">dim</a></code></span> |
| class to encapsulate pairs of base dimensions and <span class="underline"><code class="computeroutput"><a class="link" href="../boost/units/static_rational.html" title="Class template static_rational">static_rational</a></code></span> |
| exponents. The <span class="underline"><code class="computeroutput"><a class="link" href="../boost/units/make_dimension_list.html" title="Struct template make_dimension_list">make_dimension_list</a></code></span> |
| class acts as a wrapper to ensure that the resulting type is in the form of |
| a reduced dimension: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">list</span><span class="special"><</span> <span class="identifier">dim</span><span class="special"><</span> <span class="identifier">length_base_dimension</span><span class="special">,</span><span class="identifier">static_rational</span><span class="special"><</span><span class="number">1</span><span class="special">></span> <span class="special">></span> <span class="special">></span> |
| <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">length_dimension</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">list</span><span class="special"><</span> <span class="identifier">dim</span><span class="special"><</span> <span class="identifier">mass_base_dimension</span><span class="special">,</span><span class="identifier">static_rational</span><span class="special"><</span><span class="number">1</span><span class="special">></span> <span class="special">></span> <span class="special">></span> |
| <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">mass_dimension</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">list</span><span class="special"><</span> <span class="identifier">dim</span><span class="special"><</span> <span class="identifier">time_base_dimension</span><span class="special">,</span><span class="identifier">static_rational</span><span class="special"><</span><span class="number">1</span><span class="special">></span> <span class="special">></span> <span class="special">></span> |
| <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">time_dimension</span><span class="special">;</span> |
| |
| </pre> |
| <p> |
| </p> |
| <p> |
| This can also be easily accomplished using a convenience typedef provided by |
| <span class="underline"><code class="computeroutput"><a class="link" href="../boost/units/base_dimension.html" title="Class template base_dimension">base_dimension</a></code></span>: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">length_base_dimension</span><span class="special">::</span><span class="identifier">dimension_type</span> <span class="identifier">length_dimension</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">mass_base_dimension</span><span class="special">::</span><span class="identifier">dimension_type</span> <span class="identifier">mass_dimension</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">time_base_dimension</span><span class="special">::</span><span class="identifier">dimension_type</span> <span class="identifier">time_dimension</span><span class="special">;</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| so that the above code is identical to the full typelist definition. Composite |
| dimensions are similarly defined via a typelist: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">list</span><span class="special"><</span> <span class="identifier">dim</span><span class="special"><</span> <span class="identifier">length_base_dimension</span><span class="special">,</span><span class="identifier">static_rational</span><span class="special"><</span><span class="number">2</span><span class="special">></span> <span class="special">></span> <span class="special">></span> |
| <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">area_dimension</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">list</span><span class="special"><</span> <span class="identifier">dim</span><span class="special"><</span> <span class="identifier">mass_base_dimension</span><span class="special">,</span><span class="identifier">static_rational</span><span class="special"><</span><span class="number">1</span><span class="special">></span> <span class="special">>,</span> |
| <span class="identifier">dim</span><span class="special"><</span> <span class="identifier">length_base_dimension</span><span class="special">,</span><span class="identifier">static_rational</span><span class="special"><</span><span class="number">2</span><span class="special">></span> <span class="special">>,</span> |
| <span class="identifier">dim</span><span class="special"><</span> <span class="identifier">time_base_dimension</span><span class="special">,</span><span class="identifier">static_rational</span><span class="special"><-</span><span class="number">2</span><span class="special">></span> <span class="special">></span> <span class="special">></span> |
| <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">energy_dimension</span><span class="special">;</span> |
| |
| </pre> |
| <p> |
| </p> |
| <p> |
| A convenience class for composite dimensions with integer powers is also provided: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">derived_dimension</span><span class="special"><</span><span class="identifier">length_base_dimension</span><span class="special">,</span><span class="number">2</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">area_dimension</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">derived_dimension</span><span class="special"><</span><span class="identifier">mass_base_dimension</span><span class="special">,</span><span class="number">1</span><span class="special">,</span> |
| <span class="identifier">length_base_dimension</span><span class="special">,</span><span class="number">2</span><span class="special">,</span> |
| <span class="identifier">time_base_dimension</span><span class="special">,-</span><span class="number">2</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">energy_dimension</span><span class="special">;</span> |
| </pre> |
| <p> |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2003 -2008 Matthias Christian Schabel, 2007-2010 Steven |
| Watanabe<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
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