ceph/src/boost/libs/mpl/doc/tutorial/dimensional-analysis.html

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  10 <title>THE BOOST MPL LIBRARY: Dimensional Analysis</title>
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  17 <div class="section" id="dimensional-analysis">
  18 <h1><a class="toc-backref" href="./tutorial-metafunctions.html#id41" name="dimensional-analysis">Dimensional Analysis</a></h1>
  19 <p>The first rule of doing physical calculations
  20 on paper is that the numbers being manipulated don't stand alone:
  21 most quantities have attached <em>dimensions</em>, to be ignored at our
  22 peril.  As computations become more complex, keeping track of
  23 dimensions is what keeps us from inadvertently assigning a mass to
  24 what should be a length or adding acceleration to velocity — it
  25 establishes a type system for numbers.</p>
  26 <p>Manual checking of types is tedious, and as a result, it's also
  27 error-prone.  When human beings become bored, their attention
  28 wanders and they tend to make mistakes.  Doesn't type checking seem
  29 like the sort of job a computer might be good at, though?  If we
  30 could establish a framework of C++ types for dimensions and
  31 quantities, we might be able to catch errors in formulae before
  32 they cause serious problems in the real world.</p>
  33 <p>Preventing quantities with different dimensions from interoperating
  34 isn't hard; we could simply represent dimensions as classes that
  35 only work with dimensions of the same type.  What makes this
  36 problem interesting is that different dimensions <em>can</em> be combined,
  37 via multiplication or division, to produce arbitrarily complex new
  38 dimensions.  For example, take Newton's law, which relates force to
  39 mass and acceleration:</p>
  40 <blockquote>
  41 <em>F</em> = <em>ma</em></blockquote>
  42 <p>Since mass and acceleration have different dimensions, the
  43 dimensions of force must somehow capture their combination.  In
  44 fact, the dimensions of acceleration are already just such a
  45 composite, a change in velocity over time:</p>
  46 <blockquote>
  47 <em>dv</em>/<em>dt</em></blockquote>
  48 <p>Since velocity is just change in distance (<em>l</em>) over time (<em>t</em>),
  49 the fundamental dimensions of acceleration are:</p>
  50 <blockquote>
  51 (<em>l</em>/<em>t</em>)/<em>t</em>  =  <em>l</em>/<em>t</em><sup>2</sup></blockquote>
  52 <p>And indeed, acceleration is commonly measured in &quot;meters per second
  53 squared.&quot;  It follows that the dimensions of force must be:</p>
  54 <blockquote>
  55 <em>ml</em>/<em>t</em><sup>2</sup></blockquote>
  56 <!-- @litre_translator.line_offset -= 7 -->
  57 <p>and force is commonly measured in kg(m/s<sup>2</sup>), or
  58 &quot;kilogram-meters per second squared.&quot;  When multiplying quantities
  59 of mass and acceleration, we multiply their dimensions as well and
  60 carry the result along, which helps us to ensure that the result is
  61 meaningful.  The formal name for this bookkeeping is <strong>dimensional
  62 analysis</strong>, and our next task will be to implement its rules in the C++
  63 type system.  John Barton and Lee Nackman were the first to show
  64 how to do this in their seminal book, <em>Scientific and Engineering
  65 C++</em> <a class="citation-reference" href="#bn94" id="id5" name="id5">[BN94]</a>.  We will recast their approach here in
  66 metaprogramming terms.</p>
  67 <table class="citation" frame="void" id="bn94" rules="none">
  68 <colgroup><col class="label" /><col /></colgroup>
  69 <tbody valign="top">
  70 <tr><td class="label"><a class="fn-backref" href="#id5" name="bn94">[BN94]</a></td><td>John J. Barton and Lee R. Nackman. <em>Scientific and
  71 Engineering C++: an Introduction with Advanced Techniques and
  72 Examples.</em> Reading, MA: Addison Wesley. ISBN
  73 0-201-53393-6. 1994.</td></tr>
  74 </tbody>
  75 </table>
  76 <ul class="toc simple" id="outline">
  77 <li><a class="reference" href="./representing-dimensions.html" id="id42" name="id42">Representing Dimensions</a></li>
  78 <li><a class="reference" href="./representing-quantities.html" id="id43" name="id43">Representing Quantities</a></li>
  79 <li><a class="reference" href="./implementing-addition-and.html" id="id44" name="id44">Implementing Addition and Subtraction</a></li>
  80 <li><a class="reference" href="./implementing.html" id="id45" name="id45">Implementing Multiplication</a></li>
  81 <li><a class="reference" href="./implementing-division.html" id="id46" name="id46">Implementing Division</a></li>
  82 </ul>
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