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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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