1<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> 2<html> 3<head> 4<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 5<title>Dimensional Analysis</title> 6<link rel="stylesheet" href="../../../doc/src/boostbook.css" type="text/css"> 7<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 8<link rel="home" href="../index.html" title="The Boost C++ Libraries BoostBook Documentation Subset"> 9<link rel="up" href="../boost_units.html" title="Chapter 43. Boost.Units 1.1.0"> 10<link rel="prev" href="Quick_Start.html" title="Quick Start"> 11<link rel="next" href="Units.html" title="Units"> 12</head> 13<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 14<table cellpadding="2" width="100%"><tr> 15<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../boost.png"></td> 16<td align="center"><a href="../../../index.html">Home</a></td> 17<td align="center"><a href="../../../libs/libraries.htm">Libraries</a></td> 18<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 19<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 20<td align="center"><a href="../../../more/index.htm">More</a></td> 21</tr></table> 22<hr> 23<div class="spirit-nav"> 24<a accesskey="p" href="Quick_Start.html"><img src="../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../boost_units.html"><img src="../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="Units.html"><img src="../../../doc/src/images/next.png" alt="Next"></a> 25</div> 26<div class="section"> 27<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 28<a name="boost_units.Dimensional_Analysis"></a><a class="link" href="Dimensional_Analysis.html" title="Dimensional Analysis">Dimensional Analysis</a> 29</h2></div></div></div> 30<p> 31 The concept of <a href="http://en.wikipedia.org/wiki/Dimensional_analysis" target="_top">dimensional 32 analysis</a> is normally presented early on in introductory physics and 33 engineering classes as a means of determining the correctness of an equation 34 or computation by propagating the physical measurement <a href="http://en.wikipedia.org/wiki/Units_of_measurement" target="_top">units</a> 35 of various quantities through the equation along with their numerical values. 36 There are a number of standard unit systems in common use, the most prominent 37 of which is the <a href="http://en.wikipedia.org/wiki/SI_units" target="_top">Systeme 38 International</a> (also known as SI or MKS (meter-kilogram-second), which 39 was a metric predecessor to the SI system named for three of the base units 40 on which the system is based). The SI is the only official international standard 41 unit system and is widely utilized in science and engineering. Other common 42 systems include the <a href="http://en.wikipedia.org/wiki/Cgs_units" target="_top">CGS</a> 43 (centimeter-gram-second) system and the <a href="http://en.wikipedia.org/wiki/English_units" target="_top">English</a> 44 system still in use in some problem domains in the United States and elsewhere. 45 In physics, there also exist a number of other systems that are in common use 46 in specialized subdisciplines. These are collectively referred to as <a href="http://en.wikipedia.org/wiki/Natural_units" target="_top">natural units</a>. When 47 quantities representing different measurables are combined, dimensional analysis 48 provides the means of assessing the consistency of the resulting calculation. 49 For example, the sum of two lengths is also a length, while the product of 50 two lengths is an area, and the sum of a length and an area is undefined. The 51 fact that the arguments to many functions (such as exp, log, etc...) must be 52 dimensionless quantities can be easily demonstrated by examining their series 53 expansions in the context of dimensional analysis. This library facilitates 54 the enforcement of this type of restriction in code involving dimensioned quantities 55 where appropriate. 56 </p> 57<p> 58 In the following discussion we view dimensional analysis as an abstraction 59 in which an arbitrary set of <a href="http://en.wikipedia.org/wiki/Fundamental_units" target="_top">units</a> 60 obey the rules of a specific algebra. We will refer to a pair of a base dimension 61 and a rational exponent as a <span class="bold"><strong>fundamental dimension</strong></span>, 62 and a list composed of an arbitrary number of fundamental dimensions as a 63 <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 64 <span class="inlinemediaobject"><img src="../../../libs/units/images/form_3.png" alt="form_3"></span>. 65 </p> 66<p> 67 Composite dimensions obey the algebraic rules for dimensional analysis. In 68 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: 69 </p> 70<p> 71 <span class="inlinemediaobject"><img src="../../../libs/units/images/form_8.png" alt="form_8"></span> 72 </p> 73<p> 74 Users of a dimensional analysis library should be able to specify an arbitrary 75 list of base dimensions to produce a composite dimension. This potentially 76 includes repeated tags. For example, it should be possible to express energy 77 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 78 aggregate exponents of 1, 2, and -2, respectively. In order to be able to perform 79 computations on arbitrary sets of dimensions, all composite dimensions must 80 be reducible to an unambiguous final composite dimension, which we will refer 81 to as a <span class="bold"><strong>reduced dimension</strong></span>, for which 82 </p> 83<div class="orderedlist"><ol class="orderedlist" type="1"> 84<li class="listitem"> 85 fundamental dimensions are consistently ordered 86 </li> 87<li class="listitem"> 88 dimensions with zero exponent are elided. Note that reduced dimensions 89 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 90 dimension, but may have fewer. 91 </li> 92</ol></div> 93<p> 94 In our implementation, base dimensions are associated with tag types. As we 95 will ultimately represent composite dimensions as typelists, we must provide 96 some mechanism for sorting base dimension tags in order to make it possible 97 to convert an arbitrary composite dimension into a reduced dimension. For this 98 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 99 (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>) 100 uses the curiously recurring template pattern (CRTP) technique to ensure that 101 ordinals specified for base dimensions are unique: 102 </p> 103<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> 104</pre> 105<p> 106 With this, we can define the base dimensions for length, mass, and time as: 107 </p> 108<p> 109</p> 110<pre class="programlisting"><span class="comment">/// base dimension of length</span> 111<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> 112<span class="comment">/// base dimension of mass</span> 113<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> 114<span class="comment">/// base dimension of time</span> 115<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> 116</pre> 117<p> 118 </p> 119<p> 120 It is important to note that the choice of order is completely arbitrary as 121 long as each tag has a unique enumerable value; non-unique ordinals are flagged 122 as errors at compile-time. Negative ordinals are reserved for use by the library. 123 To define composite dimensions corresponding to the base dimensions, we simply 124 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> 125 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> 126 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> 127 class acts as a wrapper to ensure that the resulting type is in the form of 128 a reduced dimension: 129 </p> 130<p> 131</p> 132<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> 133 <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> 134<span class="special">>::</span><span class="identifier">type</span> <span class="identifier">length_dimension</span><span class="special">;</span> 135 136<span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> 137 <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> 138<span class="special">>::</span><span class="identifier">type</span> <span class="identifier">mass_dimension</span><span class="special">;</span> 139 140<span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> 141 <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> 142<span class="special">>::</span><span class="identifier">type</span> <span class="identifier">time_dimension</span><span class="special">;</span> 143</pre> 144<p> 145 </p> 146<p> 147 This can also be easily accomplished using a convenience typedef provided by 148 <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>: 149 </p> 150<p> 151</p> 152<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> 153<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> 154<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> 155</pre> 156<p> 157 </p> 158<p> 159 so that the above code is identical to the full typelist definition. Composite 160 dimensions are similarly defined via a typelist: 161 </p> 162<p> 163</p> 164<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> 165 <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> 166<span class="special">>::</span><span class="identifier">type</span> <span class="identifier">area_dimension</span><span class="special">;</span> 167 168<span class="keyword">typedef</span> <span class="identifier">make_dimension_list</span><span class="special"><</span> 169 <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> 170 <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> 171 <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> 172<span class="special">>::</span><span class="identifier">type</span> <span class="identifier">energy_dimension</span><span class="special">;</span> 173</pre> 174<p> 175 </p> 176<p> 177 A convenience class for composite dimensions with integer powers is also provided: 178 </p> 179<p> 180</p> 181<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> 182<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> 183 <span class="identifier">length_base_dimension</span><span class="special">,</span><span class="number">2</span><span class="special">,</span> 184 <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> 185</pre> 186<p> 187 </p> 188</div> 189<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 190<td align="left"></td> 191<td align="right"><div class="copyright-footer">Copyright © 2003-2008 Matthias Christian Schabel<br>Copyright © 2007-2010 Steven 192 Watanabe<p> 193 Distributed under the Boost Software License, Version 1.0. (See accompanying 194 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>) 195 </p> 196</div></td> 197</tr></table> 198<hr> 199<div class="spirit-nav"> 200<a accesskey="p" href="Quick_Start.html"><img src="../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../boost_units.html"><img src="../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="Units.html"><img src="../../../doc/src/images/next.png" alt="Next"></a> 201</div> 202</body> 203</html> 204
