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| 10 | <title>THE BOOST MPL LIBRARY: Representing Dimensions</title> |
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| 16 | </tr></table><div class="header-separator"></div> |
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| 17 | <div class="section" id="representing-dimensions"> |
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| 18 | <h1><a class="toc-backref" href="./dimensional-analysis.html#id42" name="representing-dimensions">Representing Dimensions</a></h1> |
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| 19 | <p>An international standard called <em>Système |
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| 20 | International d'Unites</em> (SI), breaks every quantity down into a |
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| 21 | combination of the dimensions <em>mass</em>, <em>length</em> (or <em>position</em>), |
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| 22 | <em>time</em>, <em>charge</em>, <em>temperature</em>, <em>intensity</em>, and <em>angle</em>. To be |
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| 23 | reasonably general, our system would have to be able to |
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| 24 | represent seven or more fundamental dimensions. It also needs |
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| 25 | the ability to represent composite dimensions that, like <em>force</em>, |
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| 26 | are built through multiplication or division of the fundamental |
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| 27 | ones.</p> |
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| 28 | <p>In general, a composite dimension is the product of powers of |
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| 29 | fundamental dimensions. <a class="footnote-reference" href="#divisor" id="id6" name="id6">[1]</a> If we were going to represent |
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| 30 | these powers for manipulation at runtime, we could use an array of |
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| 31 | seven <tt class="literal"><span class="pre">int</span></tt>s, with each position in the array holding the power |
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| 32 | of a different fundamental dimension:</p> |
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| 33 | <pre class="literal-block"> |
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| 34 | typedef int dimension[7]; // m l t ... |
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| 35 | dimension const mass = {1, 0, 0, 0, 0, 0, 0}; |
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| 36 | dimension const length = {0, 1, 0, 0, 0, 0, 0}; |
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| 37 | dimension const time = {0, 0, 1, 0, 0, 0, 0}; |
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| 38 | ... |
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| 39 | </pre> |
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| 40 | <table class="footnote" frame="void" id="divisor" rules="none"> |
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| 41 | <colgroup><col class="label" /><col /></colgroup> |
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| 42 | <tbody valign="top"> |
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| 43 | <tr><td class="label"><a class="fn-backref" href="#id6" name="divisor">[1]</a></td><td>Divisors just contribute negative exponents, since |
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| 44 | 1/<em>x</em> = <em>x</em><sup>-1</sup>.</td></tr> |
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| 45 | </tbody> |
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| 46 | </table> |
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| 47 | <p>In that representation, force would be:</p> |
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| 48 | <pre class="literal-block"> |
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| 49 | dimension const force = {1, 1, -2, 0, 0, 0, 0}; |
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| 50 | </pre> |
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| 51 | <!-- @compile(2) --> |
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| 52 | <!-- @litre_translator.line_offset -= 7 --> |
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| 53 | <p>that is, <em>mlt</em><sup>-2</sup>. However, if we want to get dimensions into the |
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| 54 | type system, these arrays won't do the trick: they're all |
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| 55 | the same type! Instead, we need types that <em>themselves</em> represent |
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| 56 | sequences of numbers, so that two masses have the same type and a |
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| 57 | mass is a different type from a length.</p> |
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| 58 | <p>Fortunately, the MPL provides us with a collection of <strong>type |
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| 59 | sequences</strong>. For example, we can build a sequence of the built-in |
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| 60 | signed integral types this way:</p> |
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| 61 | <pre class="literal-block"> |
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| 62 | #include <boost/mpl/vector.hpp> |
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| 63 | |
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| 64 | typedef boost::mpl::vector< |
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| 65 | signed char, short, int, long> signed_types; |
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| 66 | </pre> |
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| 67 | <p>How can we use a type sequence to represent numbers? Just as |
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| 68 | numerical metafunctions pass and return wrapper <em>types</em> having a |
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| 69 | nested <tt class="literal"><span class="pre">::value</span></tt>, so numerical sequences are really sequences of |
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| 70 | wrapper types (another example of polymorphism). To make this sort |
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| 71 | of thing easier, MPL supplies the <tt class="literal"><span class="pre">int_<N></span></tt> class template, which |
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| 72 | presents its integral argument as a nested <tt class="literal"><span class="pre">::value</span></tt>:</p> |
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| 73 | <pre class="literal-block"> |
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| 74 | #include <boost/mpl/int.hpp> |
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| 75 | |
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| 76 | namespace mpl = boost::mpl; // namespace alias |
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| 77 | static int const five = mpl::int_<5>::value; |
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| 78 | </pre> |
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| 79 | <div class="sidebar"> |
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| 80 | <p class="sidebar-title first">Namespace Aliases</p> |
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| 81 | <div class="line-block"> |
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| 82 | <div class="line"><tt class="literal"><span class="pre">namespace</span></tt> <em>alias</em> <tt class="literal"><span class="pre">=</span></tt> <em>namespace-name</em><tt class="literal"><span class="pre">;</span></tt></div> |
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| 83 | </div> |
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| 84 | <p>declares <em>alias</em> to be a synonym for <em>namespace-name</em>. Many |
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| 85 | examples in this book will use <tt class="literal"><span class="pre">mpl::</span></tt> to indicate |
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| 86 | <tt class="literal"><span class="pre">boost::mpl::</span></tt>, but will omit the alias that makes it legal |
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| 87 | C++.</p> |
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| 88 | </div> |
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| 89 | <!-- @ignore() # nonsense isn't worth testing |
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| 90 | prefix +=[''' |
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| 91 | #include <boost/mpl/int.hpp> |
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| 92 | #include <boost/mpl/vector.hpp> |
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| 93 | '''] --> |
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| 94 | <p>In fact, the library contains a whole suite of integral constant |
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| 95 | wrappers such as <tt class="literal"><span class="pre">long_</span></tt> and <tt class="literal"><span class="pre">bool_</span></tt>, each one wrapping a |
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| 96 | different type of integral constant within a class template.</p> |
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| 97 | <p>Now we can build our fundamental dimensions:</p> |
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| 98 | <pre class="literal-block"> |
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| 99 | typedef mpl::vector< |
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| 100 | mpl::int_<1>, mpl::int_<0>, mpl::int_<0>, mpl::int_<0> |
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| 101 | , mpl::int_<0>, mpl::int_<0>, mpl::int_<0> |
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| 102 | > mass; |
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| 103 | |
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| 104 | typedef mpl::vector< |
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| 105 | mpl::int_<0>, mpl::int_<1>, mpl::int_<0>, mpl::int_<0> |
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| 106 | , mpl::int_<0>, mpl::int_<0>, mpl::int_<0> |
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| 107 | > length; |
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| 108 | ... |
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| 109 | </pre> |
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| 110 | <!-- @ # We explained about the implicit namespace alias above |
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| 111 | prefix.append(""" |
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| 112 | namespace boost{namespace mpl {}} |
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| 113 | namespace mpl = boost::mpl; |
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| 114 | """) |
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| 115 | compile('all') --> |
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| 116 | <p>Whew! That's going to get tiring pretty quickly. Worse, it's hard |
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| 117 | to read and verify: The essential information, the powers of each |
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| 118 | fundamental dimension, is buried in repetitive syntactic "noise." |
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| 119 | Accordingly, MPL supplies <strong>integral sequence wrappers</strong> that allow |
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| 120 | us to write:</p> |
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| 121 | <pre class="literal-block"> |
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| 122 | #include <boost/mpl/vector_c.hpp> |
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| 123 | |
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| 124 | typedef mpl::vector_c<int,1,0,0,0,0,0,0> mass; |
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| 125 | typedef mpl::vector_c<int,0,1,0,0,0,0,0> length; // or position |
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| 126 | typedef mpl::vector_c<int,0,0,1,0,0,0,0> time; |
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| 127 | typedef mpl::vector_c<int,0,0,0,1,0,0,0> charge; |
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| 128 | typedef mpl::vector_c<int,0,0,0,0,1,0,0> temperature; |
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| 129 | typedef mpl::vector_c<int,0,0,0,0,0,1,0> intensity; |
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| 130 | typedef mpl::vector_c<int,0,0,0,0,0,0,1> angle; |
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| 131 | </pre> |
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| 132 | <p>Even though they have different types, you can think of these |
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| 133 | <tt class="literal"><span class="pre">mpl::vector_c</span></tt> specializations as being equivalent to the more |
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| 134 | verbose versions above that use <tt class="literal"><span class="pre">mpl::vector</span></tt>.</p> |
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| 135 | <p>If we want, we can also define a few composite dimensions:</p> |
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| 136 | <pre class="literal-block"> |
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| 137 | // base dimension: m l t ... |
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| 138 | typedef mpl::vector_c<int,0,1,-1,0,0,0,0> velocity; // l/t |
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| 139 | typedef mpl::vector_c<int,0,1,-2,0,0,0,0> acceleration; // l/(t<sup>2</sup>) |
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| 140 | typedef mpl::vector_c<int,1,1,-1,0,0,0,0> momentum; // ml/t |
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| 141 | typedef mpl::vector_c<int,1,1,-2,0,0,0,0> force; // ml/(t<sup>2</sup>) |
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| 142 | </pre> |
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| 143 | <p>And, incidentally, the dimensions of scalars (like pi) can be |
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| 144 | described as:</p> |
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| 145 | <pre class="literal-block"> |
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| 146 | typedef mpl::vector_c<int,0,0,0,0,0,0,0> scalar; |
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| 147 | </pre> |
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| 148 | <!-- @stack[0].replace('hpp>', 'hpp>\nnamespace {') |
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| 149 | stack[0].append('}') |
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| 150 | compile('all', pop = None) --> |
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