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