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<div class="section" id="representing-dimensions">
<h1><a class="toc-backref" href="./dimensional-analysis.html#id42" name="representing-dimensions">Representing Dimensions</a></h1>
<p>An international standard called <em>Système
International d'Unites</em> (SI), breaks every quantity down into a
combination of the dimensions <em>mass</em>, <em>length</em> (or <em>position</em>),
<em>time</em>, <em>charge</em>, <em>temperature</em>, <em>intensity</em>, and <em>angle</em>.  To be
reasonably general, our system would have to be able to
represent seven or more fundamental dimensions.  It also needs
the ability to represent composite dimensions that, like <em>force</em>,
are built through multiplication or division of the fundamental
ones.</p>
<p>In general, a composite dimension is the product of powers of
fundamental dimensions. <a class="footnote-reference" href="#divisor" id="id6" name="id6">[1]</a>  If we were going to represent
these powers for manipulation at runtime, we could use an array of
seven <tt class="literal"><span class="pre">int</span></tt>s, with each position in the array holding the power
of a different fundamental dimension:</p>
<pre class="literal-block">
typedef int dimension[7]; // m  l  t  ...
dimension const mass      = {1, 0, 0, 0, 0, 0, 0};
dimension const length    = {0, 1, 0, 0, 0, 0, 0};
dimension const time      = {0, 0, 1, 0, 0, 0, 0};
...
</pre>
<table class="footnote" frame="void" id="divisor" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id6" name="divisor">[1]</a></td><td>Divisors just contribute negative exponents, since
1/<em>x</em> = <em>x</em><sup>-1</sup>.</td></tr>
</tbody>
</table>
<p>In that representation, force would be:</p>
<pre class="literal-block">
dimension const force  = {1, 1, -2, 0, 0, 0, 0};
</pre>
<!-- @compile(2) -->
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<p>that is, <em>mlt</em><sup>-2</sup>.  However, if we want to get dimensions into the
type system, these arrays won't do the trick: they're all
the same type!  Instead, we need types that <em>themselves</em> represent
sequences of numbers, so that two masses have the same type and a
mass is a different type from a length.</p>
<p>Fortunately, the MPL provides us with a collection of <strong>type
sequences</strong>.  For example, we can build a sequence of the built-in
signed integral types this way:</p>
<pre class="literal-block">
#include &lt;boost/mpl/vector.hpp&gt;

typedef boost::mpl::vector&lt;
     signed char, short, int, long&gt; signed_types;
</pre>
<p>How can we use a type sequence to represent numbers?  Just as
numerical metafunctions pass and return wrapper <em>types</em> having a
nested <tt class="literal"><span class="pre">::value</span></tt>, so numerical sequences are really sequences of
wrapper types (another example of polymorphism).  To make this sort
of thing easier, MPL supplies the <tt class="literal"><span class="pre">int_&lt;N&gt;</span></tt> class template, which
presents its integral argument as a nested <tt class="literal"><span class="pre">::value</span></tt>:</p>
<pre class="literal-block">
#include &lt;boost/mpl/int.hpp&gt;

namespace mpl = boost::mpl; // namespace alias
static int const five = mpl::int_&lt;5&gt;::value;
</pre>
<div class="sidebar">
<p class="sidebar-title first">Namespace Aliases</p>
<div class="line-block">
<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>
</div>
<p>declares <em>alias</em> to be a synonym for <em>namespace-name</em>.  Many
examples in this book will use <tt class="literal"><span class="pre">mpl::</span></tt> to indicate
<tt class="literal"><span class="pre">boost::mpl::</span></tt>, but will omit the alias that makes it legal
C++.</p>
</div>
<!-- @ignore() # nonsense isn't worth testing
prefix +=['''
    #include <boost/mpl/int.hpp>
    #include <boost/mpl/vector.hpp>
'''] -->
<p>In fact, the library contains a whole suite of integral constant
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
different type of integral constant within a class template.</p>
<p>Now we can build our fundamental dimensions:</p>
<pre class="literal-block">
typedef mpl::vector&lt;
   mpl::int_&lt;1&gt;, mpl::int_&lt;0&gt;, mpl::int_&lt;0&gt;, mpl::int_&lt;0&gt;
 , mpl::int_&lt;0&gt;, mpl::int_&lt;0&gt;, mpl::int_&lt;0&gt;
&gt; mass;

typedef mpl::vector&lt;
   mpl::int_&lt;0&gt;, mpl::int_&lt;1&gt;, mpl::int_&lt;0&gt;, mpl::int_&lt;0&gt;
 , mpl::int_&lt;0&gt;, mpl::int_&lt;0&gt;, mpl::int_&lt;0&gt;
&gt; length;
...
</pre>
<!-- @ # We explained about the implicit namespace alias above
prefix.append("""
namespace boost{namespace mpl {}}
namespace mpl = boost::mpl;
""")
compile('all') -->
<p>Whew!  That's going to get tiring pretty quickly.  Worse, it's hard
to read and verify:  The essential information, the powers of each
fundamental dimension, is buried in repetitive syntactic &quot;noise.&quot;
Accordingly, MPL supplies <strong>integral sequence wrappers</strong> that allow
us to write:</p>
<pre class="literal-block">
#include &lt;boost/mpl/vector_c.hpp&gt;

typedef mpl::vector_c&lt;int,1,0,0,0,0,0,0&gt; mass;
typedef mpl::vector_c&lt;int,0,1,0,0,0,0,0&gt; length; // or position 
typedef mpl::vector_c&lt;int,0,0,1,0,0,0,0&gt; time;
typedef mpl::vector_c&lt;int,0,0,0,1,0,0,0&gt; charge;
typedef mpl::vector_c&lt;int,0,0,0,0,1,0,0&gt; temperature;
typedef mpl::vector_c&lt;int,0,0,0,0,0,1,0&gt; intensity;
typedef mpl::vector_c&lt;int,0,0,0,0,0,0,1&gt; angle;
</pre>
<p>Even though they have different types, you can think of these
<tt class="literal"><span class="pre">mpl::vector_c</span></tt> specializations as being equivalent to the more
verbose versions above that use <tt class="literal"><span class="pre">mpl::vector</span></tt>.</p>
<p>If we want, we can also define a few composite dimensions:</p>
<pre class="literal-block">
// base dimension:        m l  t ... 
typedef mpl::vector_c&lt;int,0,1,-1,0,0,0,0&gt; velocity;     // l/t
typedef mpl::vector_c&lt;int,0,1,-2,0,0,0,0&gt; acceleration; // l/(t<sup>2</sup>)
typedef mpl::vector_c&lt;int,1,1,-1,0,0,0,0&gt; momentum;     // ml/t
typedef mpl::vector_c&lt;int,1,1,-2,0,0,0,0&gt; force;        // ml/(t<sup>2</sup>)
</pre>
<p>And, incidentally, the dimensions of scalars (like pi) can be
described as:</p>
<pre class="literal-block">
typedef mpl::vector_c&lt;int,0,0,0,0,0,0,0&gt; scalar;
</pre>
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