89 lines
6.8 KiB
HTML
89 lines
6.8 KiB
HTML
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<!-- Copyright Aleksey Gurtovoy 2006. Distributed under the Boost -->
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<title>THE BOOST MPL LIBRARY: Dimensional Analysis</title>
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<table class="header"><tr class="header"><td class="header-group navigation-bar"><span class="navigation-group"><a href="./tutorial-metafunctions.html" class="navigation-link">Prev</a> <a href="./representing-dimensions.html" class="navigation-link">Next</a></span><span class="navigation-group-separator"> | </span><span class="navigation-group">Back <a href="./higher-order.html" class="navigation-link">Along</a></span><span class="navigation-group-separator"> | </span><span class="navigation-group"><a href="./tutorial-metafunctions.html" class="navigation-link">Up</a> <a href="../index.html" class="navigation-link">Home</a></span><span class="navigation-group-separator"> | </span><span class="navigation-group"><a href="./tutorial_toc.html" class="navigation-link">Full TOC</a></span></td>
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<td class="header-group page-location"><a href="../index.html" class="navigation-link">Front Page</a> / <a href="./tutorial-metafunctions.html" class="navigation-link">Tutorial: Metafunctions and Higher-Order Metaprogramming</a> / <a href="./dimensional-analysis.html" class="navigation-link">Dimensional Analysis</a></td>
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</tr></table><div class="header-separator"></div>
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<div class="section" id="dimensional-analysis">
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<h1><a class="toc-backref" href="./tutorial-metafunctions.html#id41" name="dimensional-analysis">Dimensional Analysis</a></h1>
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<p>The first rule of doing physical calculations
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on paper is that the numbers being manipulated don't stand alone:
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most quantities have attached <em>dimensions</em>, to be ignored at our
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peril. As computations become more complex, keeping track of
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dimensions is what keeps us from inadvertently assigning a mass to
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what should be a length or adding acceleration to velocity — it
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establishes a type system for numbers.</p>
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<p>Manual checking of types is tedious, and as a result, it's also
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error-prone. When human beings become bored, their attention
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wanders and they tend to make mistakes. Doesn't type checking seem
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like the sort of job a computer might be good at, though? If we
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could establish a framework of C++ types for dimensions and
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quantities, we might be able to catch errors in formulae before
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they cause serious problems in the real world.</p>
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<p>Preventing quantities with different dimensions from interoperating
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isn't hard; we could simply represent dimensions as classes that
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only work with dimensions of the same type. What makes this
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problem interesting is that different dimensions <em>can</em> be combined,
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via multiplication or division, to produce arbitrarily complex new
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dimensions. For example, take Newton's law, which relates force to
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mass and acceleration:</p>
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<blockquote>
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<em>F</em> = <em>ma</em></blockquote>
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<p>Since mass and acceleration have different dimensions, the
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dimensions of force must somehow capture their combination. In
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fact, the dimensions of acceleration are already just such a
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composite, a change in velocity over time:</p>
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<blockquote>
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<em>dv</em>/<em>dt</em></blockquote>
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<p>Since velocity is just change in distance (<em>l</em>) over time (<em>t</em>),
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the fundamental dimensions of acceleration are:</p>
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<blockquote>
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(<em>l</em>/<em>t</em>)/<em>t</em> = <em>l</em>/<em>t</em><sup>2</sup></blockquote>
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<p>And indeed, acceleration is commonly measured in "meters per second
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squared." It follows that the dimensions of force must be:</p>
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<blockquote>
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<em>ml</em>/<em>t</em><sup>2</sup></blockquote>
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<!-- @litre_translator.line_offset -= 7 -->
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<p>and force is commonly measured in kg(m/s<sup>2</sup>), or
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"kilogram-meters per second squared." When multiplying quantities
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of mass and acceleration, we multiply their dimensions as well and
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carry the result along, which helps us to ensure that the result is
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meaningful. The formal name for this bookkeeping is <strong>dimensional
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analysis</strong>, and our next task will be to implement its rules in the C++
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type system. John Barton and Lee Nackman were the first to show
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how to do this in their seminal book, <em>Scientific and Engineering
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C++</em> <a class="citation-reference" href="#bn94" id="id5" name="id5">[BN94]</a>. We will recast their approach here in
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metaprogramming terms.</p>
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<table class="citation" frame="void" id="bn94" rules="none">
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<colgroup><col class="label" /><col /></colgroup>
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<tbody valign="top">
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<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
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Engineering C++: an Introduction with Advanced Techniques and
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Examples.</em> Reading, MA: Addison Wesley. ISBN
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0-201-53393-6. 1994.</td></tr>
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</tbody>
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</table>
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<ul class="toc simple" id="outline">
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<li><a class="reference" href="./representing-dimensions.html" id="id42" name="id42">Representing Dimensions</a></li>
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<li><a class="reference" href="./representing-quantities.html" id="id43" name="id43">Representing Quantities</a></li>
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<li><a class="reference" href="./implementing-addition-and.html" id="id44" name="id44">Implementing Addition and Subtraction</a></li>
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<li><a class="reference" href="./implementing.html" id="id45" name="id45">Implementing Multiplication</a></li>
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<li><a class="reference" href="./implementing-division.html" id="id46" name="id46">Implementing Division</a></li>
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</ul>
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</div>
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<table class="footer"><tr class="footer"><td class="header-group navigation-bar"><span class="navigation-group"><a href="./tutorial-metafunctions.html" class="navigation-link">Prev</a> <a href="./representing-dimensions.html" class="navigation-link">Next</a></span><span class="navigation-group-separator"> | </span><span class="navigation-group">Back <a href="./higher-order.html" class="navigation-link">Along</a></span><span class="navigation-group-separator"> | </span><span class="navigation-group"><a href="./tutorial-metafunctions.html" class="navigation-link">Up</a> <a href="../index.html" class="navigation-link">Home</a></span><span class="navigation-group-separator"> | </span><span class="navigation-group"><a href="./tutorial_toc.html" class="navigation-link">Full TOC</a></span></td>
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