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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) --> |
| <!-- @litre_translator.line_offset -= 7 --> |
| <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 <boost/mpl/vector.hpp> |
| |
| typedef boost::mpl::vector< |
| signed char, short, int, long> 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_<N></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 <boost/mpl/int.hpp> |
| |
| namespace mpl = boost::mpl; // namespace alias |
| static int const five = mpl::int_<5>::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< |
| mpl::int_<1>, mpl::int_<0>, mpl::int_<0>, mpl::int_<0> |
| , mpl::int_<0>, mpl::int_<0>, mpl::int_<0> |
| > mass; |
| |
| typedef mpl::vector< |
| mpl::int_<0>, mpl::int_<1>, mpl::int_<0>, mpl::int_<0> |
| , mpl::int_<0>, mpl::int_<0>, mpl::int_<0> |
| > 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 "noise." |
| Accordingly, MPL supplies <strong>integral sequence wrappers</strong> that allow |
| us to write:</p> |
| <pre class="literal-block"> |
| #include <boost/mpl/vector_c.hpp> |
| |
| typedef mpl::vector_c<int,1,0,0,0,0,0,0> mass; |
| typedef mpl::vector_c<int,0,1,0,0,0,0,0> length; // or position |
| typedef mpl::vector_c<int,0,0,1,0,0,0,0> time; |
| typedef mpl::vector_c<int,0,0,0,1,0,0,0> charge; |
| typedef mpl::vector_c<int,0,0,0,0,1,0,0> temperature; |
| typedef mpl::vector_c<int,0,0,0,0,0,1,0> intensity; |
| typedef mpl::vector_c<int,0,0,0,0,0,0,1> 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<int,0,1,-1,0,0,0,0> velocity; // l/t |
| typedef mpl::vector_c<int,0,1,-2,0,0,0,0> acceleration; // l/(t<sup>2</sup>) |
| typedef mpl::vector_c<int,1,1,-1,0,0,0,0> momentum; // ml/t |
| typedef mpl::vector_c<int,1,1,-2,0,0,0,0> 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<int,0,0,0,0,0,0,0> scalar; |
| </pre> |
| <!-- @stack[0].replace('hpp>', 'hpp>\nnamespace {') |
| stack[0].append('}') |
| compile('all', pop = None) --> |
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