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| 4 | <!-- Copyright Aleksey Gurtovoy 2006. Distributed under the Boost --> | |
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| 10 | <title>THE BOOST MPL LIBRARY: Dimensional Analysis</title> | |
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| 15 | <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> | |
| 16 | </tr></table><div class="header-separator"></div> | |
| 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 "meters per second | |
| 53 | squared." 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 | "kilogram-meters per second squared." 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> | |
| 83 | </div> | |
| 84 | ||
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