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<h1 id="international-system-of-quantities-isq-part-3-modeling-isq">International System of Quantities (ISQ): Part 3 - Modeling ISQ<a class="headerlink" href="#international-system-of-quantities-isq-part-3-modeling-isq" title="Permanent link">¶</a></h1>
|
||
<p>The physical units libraries on the market typically only focus on modeling one or more systems
|
||
of units. However, as we have learned, this is not the only system kind to model. Another,
|
||
and maybe even more important, is a system of quantities. The most important example here is
|
||
the International System of Quantities (ISQ) defined by ISO/IEC 80000.</p>
|
||
<p>This article continues our series about the International System of Quantities. This time, we will
|
||
learn about the main ideas behind the ISQ and describe how it can be modelled in a programming
|
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language.</p>
|
||
<!-- more -->
|
||
|
||
<h2 id="articles-from-this-series">Articles from this series<a class="headerlink" href="#articles-from-this-series" title="Permanent link">¶</a></h2>
|
||
<ul>
|
||
<li><a href="../../07/international-system-of-quantities-isq-part-1---introduction/">Part 1 - Introduction</a></li>
|
||
<li><a href="../../14/international-system-of-quantities-isq-part-2---problems-when-isq-is-not-used/">Part 2 - Problems when ISQ is not used</a></li>
|
||
<li>Part 3 - Modeling ISQ</li>
|
||
<li><a href="../../28/international-system-of-quantities-isq-part-4---implementing-isq/">Part 4 - Implementing ISQ</a></li>
|
||
<li><a href="../../../11/04/international-system-of-quantities-isq-part-5---benefits/">Part 5 - Benefits</a></li>
|
||
<li><a href="../../../11/11/international-system-of-quantities-isq-part-6---challenges/">Part 6 - Challenges</a></li>
|
||
</ul>
|
||
<h2 id="dimension-is-not-enough-to-describe-a-quantity">Dimension is not enough to describe a quantity<a class="headerlink" href="#dimension-is-not-enough-to-describe-a-quantity" title="Permanent link">¶</a></h2>
|
||
<p>Most of the products on the market are aware of physical dimensions. However, a dimension is not
|
||
enough to describe a quantity. Let's repeat briefly some of the problems described in more detail
|
||
in the previous article. For example, let's see the following implementation:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-0-1"><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">Box</span><span class="w"> </span><span class="p">{</span>
|
||
</span><span id="__span-0-2"><a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="n">area</span><span class="w"> </span><span class="n">base_</span><span class="p">;</span>
|
||
</span><span id="__span-0-3"><a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="n">length</span><span class="w"> </span><span class="n">height_</span><span class="p">;</span>
|
||
</span><span id="__span-0-4"><a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="k">public</span><span class="o">:</span>
|
||
</span><span id="__span-0-5"><a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="n">Box</span><span class="p">(</span><span class="n">length</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">length</span><span class="w"> </span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">length</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">base_</span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">w</span><span class="p">),</span><span class="w"> </span><span class="n">height_</span><span class="p">(</span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="p">{}</span>
|
||
</span><span id="__span-0-6"><a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="c1">// ...</span>
|
||
</span><span id="__span-0-7"><a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="p">};</span>
|
||
</span><span id="__span-0-8"><a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>
|
||
</span><span id="__span-0-9"><a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">Box</span><span class="w"> </span><span class="nf">my_box</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
<p>How do you like such an interface? It turns out that in most existing strongly-typed libraries
|
||
this is often the best we can do. <img alt="🥴" class="twemoji" src="https://cdn.jsdelivr.net/gh/jdecked/twemoji@15.1.0/assets/svg/1f974.svg" title=":woozy_face:" /></p>
|
||
<p>Another typical question many users ask is how to deal with <em>work</em> and <em>torque</em>.
|
||
Both of those have the same dimension but are distinct quantities.</p>
|
||
<p>A similar issue is related to figuring out what should be the result of:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-1-1"><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="k">auto</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">Hz</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">Bq</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">Bd</span><span class="p">;</span>
|
||
</span></code></pre></div>
|
||
<p>where:</p>
|
||
<ul>
|
||
<li><code>Hz</code> (hertz) - unit of <em>frequency</em>,</li>
|
||
<li><code>Bq</code> (becquerel) - unit of <em>activity</em>,</li>
|
||
<li><code>Bd</code> (baud) - unit of <em>modulation rate</em>.</li>
|
||
</ul>
|
||
<p>All of those quantities have the same dimension, namely <span class="arithmatex">\(\mathsf{T}^{-1}\)</span>, but probably it
|
||
is not wise to allow adding, subtracting, or comparing them, as they describe vastly different
|
||
physical properties.</p>
|
||
<p>If the above example seems too abstract, let's consider Gy (gray - unit of <em>absorbed dose</em>)
|
||
and Sv (sievert - unit of <em>dose equivalent</em>), or radian and steradian. All of those quantities
|
||
have the same dimensions.</p>
|
||
<p>Another example here is <em>fuel consumption</em> (<em>fuel volume</em> divided by <em>distance</em>, e.g.,
|
||
<code>6.7 l/100km</code>) and an <em>area</em>. Again, both have the same dimension <span class="arithmatex">\(\mathsf{L}^{2}\)</span>, but probably
|
||
it wouldn't be wise to allow adding, subtracting, or comparing a <em>fuel consumption</em> of a car
|
||
and the <em>area</em> of a football field. Such an operation does not have any physical sense and should
|
||
fail to compile.</p>
|
||
<p>It turns out that the above issues can't be solved correctly without proper modeling of
|
||
a <a href="../../../../../appendix/glossary/#system-of-quantities">system of quantities</a>.</p>
|
||
<h2 id="quantities-of-the-same-kind">Quantities of the same kind<a class="headerlink" href="#quantities-of-the-same-kind" title="Permanent link">¶</a></h2>
|
||
<p>As it was described in the previous article, dimension is not enough to describe a quantity.
|
||
We need a better abstraction to ensure the safety of our calculations. It turns out that
|
||
ISO/IEC 80000 comes with the answer:</p>
|
||
<div class="admonition quote">
|
||
<p class="admonition-title">ISO 80000-1:2009</p>
|
||
<ul>
|
||
<li>Quantities may be grouped together into categories of quantities that are
|
||
<strong>mutually comparable</strong>.</li>
|
||
<li>Mutually comparable quantities are called <strong>quantities of the same kind</strong>.</li>
|
||
<li>Two or more quantities <strong>cannot be added or subtracted unless they belong to the same category
|
||
of mutually comparable quantities</strong>.</li>
|
||
<li>Quantities of the <strong>same kind</strong> within a given system of quantities <strong>have the same quantity
|
||
dimension</strong>.</li>
|
||
<li>Quantities of the <strong>same dimension are not necessarily of the same kind</strong>.</li>
|
||
</ul>
|
||
</div>
|
||
<p>ISO Guide also explicitly states:</p>
|
||
<div class="admonition quote">
|
||
<p class="admonition-title">ISO Guide</p>
|
||
<p><strong>Measurement units of quantities of the same quantity dimension may be designated by the same
|
||
name and symbol even when the quantities are not of the same kind</strong>. For example,
|
||
joule per kelvin and J/K are respectively the name and symbol of both a measurement unit of
|
||
<em>heat capacity</em> and a measurement unit of <em>entropy</em>, which are generally not considered to be
|
||
quantities of the same kind. <strong>However, in some cases special measurement unit names are
|
||
restricted to be used with quantities of specific kind only</strong>. For example, the measurement
|
||
unit ‘second to the power minus one’ (1/s) is called hertz (Hz) when used for <em>frequencies</em>
|
||
and becquerel (Bq) when used for <em>activities of radionuclides</em>. As another example, the joule
|
||
(J) is used as a unit of <em>energy</em>, but never as a unit of <em>moment of force</em>,
|
||
i.e. the newton metre (N · m).</p>
|
||
</div>
|
||
<p>The above quotes from ISO provide answers to all the issues mentioned above and in the previous
|
||
article.</p>
|
||
<p>More than one quantity may be defined for the same dimension:</p>
|
||
<ul>
|
||
<li>quantities of different kinds (e.g., <em>frequency</em>, <em>modulation rate</em>, <em>activity</em>).</li>
|
||
<li>quantities of the same kind (e.g., <em>length</em>, <em>width</em>, <em>altitude</em>, <em>distance</em>, <em>radius</em>,
|
||
<em>wavelength</em>, <em>position vector</em>).</li>
|
||
</ul>
|
||
<p>Two quantities can't be added, subtracted, or compared unless they belong to
|
||
the same <a href="../../../../../appendix/glossary/#kind">kind</a>. As <em>frequency</em>, <em>activity</em>, and <em>modulation rate</em>
|
||
are of different kinds, the expression provided above should not compile.</p>
|
||
<h2 id="system-of-quantities-is-not-only-about-kinds">System of quantities is not only about kinds<a class="headerlink" href="#system-of-quantities-is-not-only-about-kinds" title="Permanent link">¶</a></h2>
|
||
<p>ISO/IEC 80000 specifies hundreds of different quantities. Plenty of various kinds are provided,
|
||
and often, each kind contains more than one quantity. It turns out that such quantities form
|
||
a hierarchy of quantities of the same kind.</p>
|
||
<p>For example, here are all quantities of the kind length provided in the ISO 80000-3:</p>
|
||
<pre class="mermaid"><code>flowchart TD
|
||
length["<b>length</b><br>[m]"]
|
||
length --- width["<b>width</b> / <b>breadth</b>"]
|
||
length --- height["<b>height</b> / <b>depth</b> / <b>altitude</b>"]
|
||
width --- thickness["<b>thickness</b>"]
|
||
width --- diameter["<b>diameter</b>"]
|
||
width --- radius["<b>radius</b>"]
|
||
length --- path_length["<b>path_length</b>"]
|
||
path_length --- distance["<b>distance</b>"]
|
||
distance --- radial_distance["<b>radial_distance</b>"]
|
||
length --- wavelength["<b>wavelength</b>"]
|
||
length --- displacement["<b>displacement</b><br>{vector}"]
|
||
displacement --- position_vector["<b>position_vector</b>"]
|
||
radius --- radius_of_curvature["<b>radius_of_curvature</b>"]</code></pre>
|
||
<p>Each of the above quantities expresses some kind of <em>length</em>, and each can be measured with meters,
|
||
which is the unit defined by the SI for quantities of <em>length</em>. However, each has different
|
||
properties, usage, and sometimes even a different character (<em>position vector</em> and <em>displacement</em>
|
||
are vector quantities).</p>
|
||
<p>Forming such a hierarchy helps us define arithmetics and conversion rules for various
|
||
quantities of the same kind.</p>
|
||
<h2 id="converting-between-quantities-of-the-same-kind">Converting between quantities of the same kind<a class="headerlink" href="#converting-between-quantities-of-the-same-kind" title="Permanent link">¶</a></h2>
|
||
<p>Based on the hierarchy above, we can define the following quantity conversion rules:</p>
|
||
<ol>
|
||
<li>
|
||
<p><strong>Implicit conversions</strong></p>
|
||
<ul>
|
||
<li>Every <em>width</em> is a <em>length</em>.</li>
|
||
<li>Every <em>radius</em> is a <em>width</em>.</li>
|
||
</ul>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-2-1"><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">));</span>
|
||
</span><span id="__span-2-2"><a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">radius</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">));</span>
|
||
</span><span id="__span-2-3"><a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">radius</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">));</span>
|
||
</span></code></pre></div>
|
||
<p>Implicit conversions are allowed on copy-initialization:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-3-1"><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">foo</span><span class="p">(</span><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q</span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-4-1"><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">42</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
|
||
</span><span id="__span-4-2"><a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">q1</span><span class="p">;</span><span class="w"> </span><span class="c1">// implicit quantity conversion</span>
|
||
</span><span id="__span-4-3"><a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="n">foo</span><span class="p">(</span><span class="n">q1</span><span class="p">);</span><span class="w"> </span><span class="c1">// implicit quantity conversion</span>
|
||
</span></code></pre></div>
|
||
</li>
|
||
<li>
|
||
<p><strong>Explicit conversions</strong></p>
|
||
<ul>
|
||
<li>Not every <em>length</em> is a <em>width</em>.</li>
|
||
<li>Not every <em>width</em> is a <em>radius</em>.</li>
|
||
</ul>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-5-1"><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">));</span>
|
||
</span><span id="__span-5-2"><a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">radius</span><span class="p">));</span>
|
||
</span><span id="__span-5-3"><a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">radius</span><span class="p">));</span>
|
||
</span><span id="__span-5-4"><a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">explicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">));</span>
|
||
</span><span id="__span-5-5"><a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">explicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">radius</span><span class="p">));</span>
|
||
</span><span id="__span-5-6"><a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">explicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">radius</span><span class="p">));</span>
|
||
</span></code></pre></div>
|
||
<p>Explicit conversions are forced by passing the quantity to a call operator of a <code>quantity_spec</code>
|
||
type or by calling <code>quantity</code>'s explicit constructor:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-6-1"><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">foo</span><span class="p">(</span><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q</span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-7-1"><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">42</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
|
||
</span><span id="__span-7-2"><a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">(</span><span class="n">q1</span><span class="p">);</span><span class="w"> </span><span class="c1">// explicit quantity conversion</span>
|
||
</span><span id="__span-7-3"><a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q3</span><span class="p">(</span><span class="n">q1</span><span class="p">);</span><span class="w"> </span><span class="c1">// direct initialization</span>
|
||
</span><span id="__span-7-4"><a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="n">foo</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">(</span><span class="n">q1</span><span class="p">));</span><span class="w"> </span><span class="c1">// explicit quantity conversion</span>
|
||
</span></code></pre></div>
|
||
</li>
|
||
<li>
|
||
<p><strong>Explicit casts</strong></p>
|
||
<ul>
|
||
<li><em>height</em> is never a <em>width</em>, and vice versa.</li>
|
||
<li>Both <em>height</em> and <em>width</em> are quantities of kind <em>length</em>.</li>
|
||
</ul>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-8-1"><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">));</span>
|
||
</span><span id="__span-8-2"><a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">explicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">));</span>
|
||
</span><span id="__span-8-3"><a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">castable</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">));</span>
|
||
</span></code></pre></div>
|
||
<p>Explicit casts are forced with a dedicated <code>quantity_cast</code> function:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-9-1"><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">foo</span><span class="p">(</span><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q</span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-10-1"><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">42</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
|
||
</span><span id="__span-10-2"><a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">quantity_cast</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="o">></span><span class="p">(</span><span class="n">q1</span><span class="p">);</span><span class="w"> </span><span class="c1">// explicit quantity cast</span>
|
||
</span><span id="__span-10-3"><a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="n">foo</span><span class="p">(</span><span class="n">quantity_cast</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="o">></span><span class="p">(</span><span class="n">q1</span><span class="p">));</span><span class="w"> </span><span class="c1">// explicit quantity cast</span>
|
||
</span></code></pre></div>
|
||
</li>
|
||
<li>
|
||
<p><strong>No conversion</strong></p>
|
||
<ul>
|
||
<li><em>time</em> has nothing in common with <em>length</em>.</li>
|
||
</ul>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-11-1"><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">));</span>
|
||
</span><span id="__span-11-2"><a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">explicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">));</span>
|
||
</span><span id="__span-11-3"><a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">castable</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">));</span>
|
||
</span></code></pre></div>
|
||
<p>Even the explicit casts will not force such a conversion:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-12-1"><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">foo</span><span class="p">(</span><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-13-1"><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">42</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">s</span><span class="p">;</span><span class="w"> </span><span class="c1">// Compile-time error</span>
|
||
</span><span id="__span-13-2"><a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="n">foo</span><span class="p">(</span><span class="n">quantity_cast</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="o">></span><span class="p">(</span><span class="mi">42</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">s</span><span class="p">));</span><span class="w"> </span><span class="c1">// Compile-time error</span>
|
||
</span></code></pre></div>
|
||
</li>
|
||
</ol>
|
||
<h2 id="comparing-adding-and-subtracting-quantities-of-the-same-kind">Comparing, adding, and subtracting quantities of the same kind<a class="headerlink" href="#comparing-adding-and-subtracting-quantities-of-the-same-kind" title="Permanent link">¶</a></h2>
|
||
<p>ISO/IEC 80000 explicitly states that <em>width</em> and <em>height</em> are quantities of the same kind,
|
||
and as such they:</p>
|
||
<ul>
|
||
<li>are mutually comparable,</li>
|
||
<li>can be added and subtracted.</li>
|
||
</ul>
|
||
<p>This means that we should be allowed to compare any quantities from the same tree (as long as
|
||
their underlying representation types are comparable):</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-14-1"><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">radius</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">));</span>
|
||
</span></code></pre></div>
|
||
<p>Also, based on our hierarchy above, the only reasonable result of <code>1 * width + 1 * height</code> is
|
||
<code>2 * length</code>, where the result of <code>length</code> is known as a <strong>common quantity</strong> type. A result
|
||
of such an equation is always the first common node in a hierarchy tree of the same kind.
|
||
For example:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-15-1"><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="k">static_assert</span><span class="p">((</span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">)).</span><span class="n">quantity_spec</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">);</span>
|
||
</span><span id="__span-15-2"><a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="k">static_assert</span><span class="p">((</span><span class="n">isq</span><span class="o">::</span><span class="n">thickness</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">radius</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">)).</span><span class="n">quantity_spec</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">);</span>
|
||
</span><span id="__span-15-3"><a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="k">static_assert</span><span class="p">((</span><span class="n">isq</span><span class="o">::</span><span class="n">distance</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">path_length</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">m</span><span class="p">)).</span><span class="n">quantity_spec</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">path_length</span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
<h2 id="modeling-a-quantity-kind">Modeling a quantity kind<a class="headerlink" href="#modeling-a-quantity-kind" title="Permanent link">¶</a></h2>
|
||
<p>In the quantities and units library, we also need an abstraction describing an entire family of
|
||
quantities of the same kind. Such quantities have not only the same dimension but also
|
||
can be expressed in the same units.</p>
|
||
<p>To annotate a quantity to represent its kind (and not just a hierarchy tree's root quantity)
|
||
we introduced a <code>kind_of<></code> specifier. For example, to express any quantity of <em>length</em>, we need
|
||
to type <code>kind_of<isq::length></code>.</p>
|
||
<div class="admonition important">
|
||
<p class="admonition-title">Important</p>
|
||
<p><code>isq::length</code> and <code>kind_of<isq::length></code> are two different things.</p>
|
||
</div>
|
||
<p>Such an entity behaves as any quantity of its kind. This means that it is implicitly
|
||
convertible to any quantity in a tree.</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-16-1"><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">));</span>
|
||
</span><span id="__span-16-2"><a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">implicitly_convertible</span><span class="p">(</span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="o">></span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">height</span><span class="p">));</span>
|
||
</span></code></pre></div>
|
||
<p>Additionally, the result of operations on quantity kinds is also a quantity kind:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-17-1"><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">same_type</span><span class="o"><</span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="o">></span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="o">></span><span class="p">,</span><span class="w"> </span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="o">>></span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
<p>However, if at least one equation's operand is not a quantity kind, the result becomes a "strong"
|
||
quantity where all the kinds are converted to the hierarchy tree's root quantities:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-18-1"><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="o">!</span><span class="n">same_type</span><span class="o"><</span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="o">></span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="p">,</span><span class="w"> </span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="o">>></span><span class="p">);</span>
|
||
</span><span id="__span-18-2"><a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">same_type</span><span class="o"><</span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="o">></span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="p">,</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">isq</span><span class="o">::</span><span class="n">time</span><span class="o">></span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
<div class="admonition info">
|
||
<p class="admonition-title">Info</p>
|
||
<p>Only a root quantity from the hierarchy tree or the one marked with <code>is_kind</code> specifier
|
||
in the <code>quantity_spec</code> definition can be put as a template parameter to the <code>kind_of</code>
|
||
specifier. For example, <code>kind_of<isq::width></code> will fail to compile. However, we can call
|
||
<code>get_kind(q)</code> to obtain a kind of any quantity:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-19-1"><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="k">static_assert</span><span class="p">(</span><span class="n">get_kind</span><span class="p">(</span><span class="n">isq</span><span class="o">::</span><span class="n">width</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="o">></span><span class="p">);</span>
|
||
</span></code></pre></div>
|
||
</div>
|
||
<h2 id="how-do-systems-of-units-benefit-from-the-isq-and-quantity-kinds">How do systems of units benefit from the ISQ and quantity kinds?<a class="headerlink" href="#how-do-systems-of-units-benefit-from-the-isq-and-quantity-kinds" title="Permanent link">¶</a></h2>
|
||
<p>Modeling a system of units is the most essential feature and a selling point of every physical
|
||
units library. Thanks to that, the library can protect users from assigning, adding, subtracting,
|
||
or comparing incompatible units and provide automated conversion factors between various compatible
|
||
units.</p>
|
||
<p>Probably all the libraries in the wild model the SI (or at least most of it), and many of them
|
||
provide support for additional units belonging to various other systems (e.g., imperial).</p>
|
||
<h3 id="systems-of-units-are-based-on-systems-of-quantities">Systems of units are based on systems of quantities<a class="headerlink" href="#systems-of-units-are-based-on-systems-of-quantities" title="Permanent link">¶</a></h3>
|
||
<p>Systems of quantities specify a set of quantities and equations relating to those quantities.
|
||
Those equations do not take any unit or a numerical representation into account at all. In order
|
||
to create a quantity, we need to add those missing pieces of information. This is where
|
||
a system of units kicks in.</p>
|
||
<p>The SI is explicitly stated to be based on the ISQ. Among others, it defines seven base units,
|
||
one for each base quantity of ISQ. In the library, this is expressed by associating a quantity
|
||
kind to a unit being defined:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-20-1"><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="kr">inline</span><span class="w"> </span><span class="k">constexpr</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">metre</span><span class="w"> </span><span class="k">final</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">named_unit</span><span class="o"><</span><span class="s">"m"</span><span class="p">,</span><span class="w"> </span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">length</span><span class="o">>></span><span class="w"> </span><span class="p">{}</span><span class="w"> </span><span class="n">metre</span><span class="p">;</span>
|
||
</span></code></pre></div>
|
||
<p>The <code>kind_of<isq::length></code> above states explicitly that this unit has an associated quantity
|
||
kind. In other words, <code>si::metre</code> (and scaled units based on it) can be used to express
|
||
the amount of any quantity of kind <em>length</em>.</p>
|
||
<div class="admonition note">
|
||
<p class="admonition-title">Note</p>
|
||
<p>For some systems of units (e.g., natural units), a unit may not have an associated quantity
|
||
type. For example, if we define the speed of light constant as <code>c = 1</code>, we can define a system
|
||
where both <em>length</em> and <em>time</em> will be measured in seconds, and <em>speed</em> will be a quantity
|
||
measured with the unit <code>one</code>. In such case, the definition will look as follows:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-21-1"><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="kr">inline</span><span class="w"> </span><span class="k">constexpr</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">second</span><span class="w"> </span><span class="k">final</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">named_unit</span><span class="o"><</span><span class="s">"s"</span><span class="o">></span><span class="w"> </span><span class="p">{}</span><span class="w"> </span><span class="n">second</span><span class="p">;</span>
|
||
</span></code></pre></div>
|
||
</div>
|
||
<h3 id="constraining-a-derived-unit-to-work-only-with-a-specific-derived-quantity">Constraining a derived unit to work only with a specific derived quantity<a class="headerlink" href="#constraining-a-derived-unit-to-work-only-with-a-specific-derived-quantity" title="Permanent link">¶</a></h3>
|
||
<p>Some derived units are valid only for specific derived quantities. For example, SI specifies
|
||
both hertz and becquerel derived units with the same unit equation <span class="arithmatex">\(s^{-1}\)</span>. However, it also
|
||
explicitly states:</p>
|
||
<div class="admonition quote">
|
||
<p class="admonition-title">SI</p>
|
||
<p>The hertz shall only be used for periodic phenomena and the becquerel shall only be used for
|
||
stochastic processes in activity referred to a radionuclide.</p>
|
||
</div>
|
||
<p>This is why it is important for the library to allow constraining such units to be used only with
|
||
a specific quantity kind:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-22-1"><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="kr">inline</span><span class="w"> </span><span class="k">constexpr</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">hertz</span><span class="w"> </span><span class="k">final</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">named_unit</span><span class="o"><</span><span class="s">"Hz"</span><span class="p">,</span><span class="w"> </span><span class="n">one</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">second</span><span class="p">,</span><span class="w"> </span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">frequency</span><span class="o">>></span><span class="w"> </span><span class="p">{}</span><span class="w"> </span><span class="n">hertz</span><span class="p">;</span>
|
||
</span><span id="__span-22-2"><a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kr">inline</span><span class="w"> </span><span class="k">constexpr</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">becquerel</span><span class="w"> </span><span class="k">final</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">named_unit</span><span class="o"><</span><span class="s">"Bq"</span><span class="p">,</span><span class="w"> </span><span class="n">one</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">second</span><span class="p">,</span><span class="w"> </span><span class="n">kind_of</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">activity</span><span class="o">>></span><span class="w"> </span><span class="p">{}</span><span class="w"> </span><span class="n">becquerel</span><span class="p">;</span>
|
||
</span></code></pre></div>
|
||
<p>With the above, <code>hertz</code> can only be used for <em>frequencies</em>, while <code>becquerel</code> should only be used
|
||
for quantities of <em>activity</em>:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-23-1"><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">frequency</span><span class="p">[</span><span class="n">Hz</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">60</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">Bq</span><span class="p">;</span><span class="w"> </span><span class="c1">// Compile-time error</span>
|
||
</span><span id="__span-23-2"><a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">activity</span><span class="p">[</span><span class="n">Hz</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Compile-time error</span>
|
||
</span><span id="__span-23-3"><a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="n">quantity</span><span class="o"><</span><span class="n">isq</span><span class="o">::</span><span class="n">frequency</span><span class="p">[</span><span class="n">Hz</span><span class="p">]</span><span class="o">></span><span class="w"> </span><span class="n">q3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">60</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">Hz</span><span class="p">;</span><span class="w"> </span><span class="c1">// OK</span>
|
||
</span><span id="__span-23-4"><a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="n">std</span><span class="o">::</span><span class="n">cout</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="n">q3</span><span class="p">.</span><span class="n">in</span><span class="p">(</span><span class="n">Bq</span><span class="p">)</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="s">"</span><span class="se">\n</span><span class="s">"</span><span class="p">;</span><span class="w"> </span><span class="c1">// Compile-time error</span>
|
||
</span></code></pre></div>
|
||
<p>We know already that quantities of different kinds can't be compared, added, and subtracted.
|
||
The following equation will not compile thanks to constraining derived units to be valid for
|
||
specific kinds only:</p>
|
||
<div class="language-cpp highlight"><pre><span></span><code><span id="__span-24-1"><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="k">auto</span><span class="w"> </span><span class="n">q</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">Hz</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">Bq</span><span class="p">;</span><span class="w"> </span><span class="c1">// Fails to compile</span>
|
||
</span></code></pre></div>
|
||
<p>All of the above features improve the safety of our library and the products that are using it.</p>
|
||
<h2 id="to-be-continued">To be continued...<a class="headerlink" href="#to-be-continued" title="Permanent link">¶</a></h2>
|
||
<p>In the next part of this series, we will present how we can implement our ISQ model in a C++
|
||
programming language and we will point out some of the first issues that stand in our way.</p>
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