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<a name="math_toolkit.catmull_rom"></a><a class="link" href="catmull_rom.html" title="Catmull-Rom Splines">Catmull-Rom Splines</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.catmull_rom.h0"></a>
<span class="phrase"><a name="math_toolkit.catmull_rom.synopsis"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.synopsis">Synopsis</a>
</h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">catmull_rom</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Point</span><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">catmull_rom</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="identifier">catmull_rom</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Point</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">points</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">num_points</span><span class="special">,</span> <span class="keyword">bool</span> <span class="identifier">closed</span> <span class="special">=</span> <span class="keyword">false</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">1</span><span class="special">/</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">2</span><span class="special">)</span>
<span class="identifier">catmull_rom</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Point</span><span class="special">&gt;&amp;&amp;</span> <span class="identifier">points</span><span class="special">,</span> <span class="keyword">bool</span> <span class="identifier">closed</span> <span class="special">=</span> <span class="keyword">false</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">1</span><span class="special">/</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">2</span><span class="special">)</span>
<span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">s</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">max_parameter</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Real</span> <span class="identifier">parameter_at_point</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">Point</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">s</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="special">}}</span>
</pre>
<h4>
<a name="math_toolkit.catmull_rom.h1"></a>
<span class="phrase"><a name="math_toolkit.catmull_rom.description"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.description">Description</a>
</h4>
<p>
Catmull-Rom splines are a family of interpolating curves which are commonly
used in computer graphics and animation. Catmull-Rom splines enjoy the following
properties:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Affine invariance: The interpolant commutes with affine transformations.
</li>
<li class="listitem">
Local support of the basis functions: This gives stability and fast evaluation.
</li>
<li class="listitem">
Smoothness
</li>
<li class="listitem">
Interpolation of control points: Many curves (such as Bezier) are <span class="bold"><strong>approximating</strong></span>, they do not pass through their control
points. This makes them more difficult to use than interpolating splines.
</li>
</ul></div>
<p>
The <code class="computeroutput"><span class="identifier">catmull_rom</span></code> class provided
by boost creates a cubic Catmull-Rom spline from an array of points in any
dimension. Since there are numerous ways to represent a point in <span class="emphasis"><em>n</em></span>-dimensional
space, the class attempts to be flexible by templating on the point type. The
requirements on the point type are discussing in more detail below, but roughly,
it must have a dereference operator defined (i.e., <code class="computeroutput"><span class="identifier">p</span><span class="special">[</span><span class="number">0</span><span class="special">]</span></code>
is not a syntax error), it must be able to be dereferenced up to <code class="computeroutput"><span class="identifier">dimension</span> <span class="special">-</span><span class="number">1</span></code>, and <code class="computeroutput"><span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span></code>
is of type <code class="computeroutput"><span class="identifier">Real</span></code>, define <code class="computeroutput"><span class="identifier">value_type</span></code>, and the free function <code class="computeroutput"><span class="identifier">size</span><span class="special">()</span></code>. These
requirements are met by <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
and <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code>. The basic usage is shown here:
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">points</span><span class="special">(</span><span class="number">4</span><span class="special">);</span>
<span class="identifier">points</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
<span class="identifier">points</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
<span class="identifier">points</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
<span class="identifier">points</span><span class="special">[</span><span class="number">3</span><span class="special">]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
<span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">points</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">points</span><span class="special">.</span><span class="identifier">size</span><span class="special">());</span>
<span class="comment">// Interpolate at s = 0.1:</span>
<span class="keyword">auto</span> <span class="identifier">point</span> <span class="special">=</span> <span class="identifier">cr</span><span class="special">(</span><span class="number">0.1</span><span class="special">);</span>
</pre>
<p>
Memory conscious programmers may enjoy using the move constructor instead:
</p>
<pre class="programlisting"><span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">points</span><span class="special">));</span>
</pre>
<p>
The spline can be either open or <span class="bold"><strong>closed</strong></span>, closed
meaning that there is some <span class="emphasis"><em>t</em></span> such that <span class="emphasis"><em>P(t)
= P(0)</em></span>. The default is open, but this can be easily changed:
</p>
<pre class="programlisting"><span class="comment">// closed = true</span>
<span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">points</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">points</span><span class="special">.</span><span class="identifier">size</span><span class="special">(),</span> <span class="keyword">true</span><span class="special">);</span>
</pre>
<p>
Inside <code class="computeroutput"><span class="identifier">catmull_rom</span></code>, the Catmull-Rom
curve is represented as closed. This is because an open Catmull-Rom curve is
<span class="bold"><strong>implicitly closed</strong></span>, but the closing point is
the zero vector. There is no reason to suppose that the zero vector is a better
closing point than the endpoint (or any other point, for that matter), so traditionally
Catmull-Rom splines leave the segment between the first and second point undefined,
as well as the segment between the second-to-last and last point. We find this
property of the traditional implementation of Catmull-Rom splines annoying
and confusing to the user. Hence internally, we close the curve so that the
first and last segments are defined. Of course, this causes the <span class="bold"><strong>tangent</strong></span>
vectors to the first and last points to be bizarre. This is a "pick your
poison" design decision-either the curve cannot interpolate in its first
and last segments, or the tangents along the first and last segments are meaningless.
</p>
<p>
Since the routine internally represents the curve as closed, a question arises:
Why does the user have to specify if the curve is open or closed? The answer
is that the parameterization is chosen by the routine, so it is of interest
to the user to understand the values where a meaningful result is returned.
</p>
<pre class="programlisting"><span class="identifier">Real</span> <span class="identifier">max_s</span> <span class="special">=</span> <span class="identifier">cr</span><span class="special">.</span><span class="identifier">max_parameter</span><span class="special">();</span>
</pre>
<p>
If you attempt to interpolate for <code class="computeroutput"><span class="identifier">s</span>
<span class="special">&gt;</span> <span class="identifier">max_s</span></code>,
an exception is thrown. If the curve is closed, then <code class="computeroutput"><span class="identifier">cr</span><span class="special">(</span><span class="identifier">max_s</span><span class="special">)</span>
<span class="special">=</span> <span class="identifier">p0</span></code>,
where <code class="computeroutput"><span class="identifier">p0</span></code> is the first point
on the curve. If the curve is open, then <code class="computeroutput"><span class="identifier">cr</span><span class="special">(</span><span class="identifier">max_s</span><span class="special">)</span>
<span class="special">=</span> <span class="identifier">pf</span></code>,
where <code class="computeroutput"><span class="identifier">pf</span></code> is the final point
on the curve.
</p>
<p>
The Catmull-Rom curve admits an infinite number of parameterizations. The default
parameterization of the <code class="computeroutput"><span class="identifier">catmull_rom</span></code>
class is the so-called <span class="bold"><strong>centripedal</strong></span> parameterization.
This parameterization has been shown to be the only parameterization that does
not form cusps or self-intersections within segments. However, for advanced
users, other parameterizations can be chosen using the <span class="emphasis"><em>alpha</em></span>
parameter:
</p>
<pre class="programlisting"><span class="comment">// alpha = 1 is the "chordal" parameterization.</span>
<span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">points</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">points</span><span class="special">.</span><span class="identifier">size</span><span class="special">(),</span> <span class="keyword">false</span><span class="special">,</span> <span class="number">1.0</span><span class="special">);</span>
</pre>
<p>
Finally, the tangent vector to any point of the curve can be computed via
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">s</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span>
<span class="identifier">Point</span> <span class="identifier">tangent</span> <span class="special">=</span> <span class="identifier">cr</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="identifier">s</span><span class="special">);</span>
</pre>
<p>
Since the magnitude of the tangent vector is dependent on the parameterization,
it is not as meaningful as (say) arc-length parameterization. However, its
direction is meaningful, so the user may wish to normalize this result.
</p>
<h4>
<a name="math_toolkit.catmull_rom.h2"></a>
<span class="phrase"><a name="math_toolkit.catmull_rom.examples"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.examples">Examples</a>
</h4>
<h4>
<a name="math_toolkit.catmull_rom.h3"></a>
<span class="phrase"><a name="math_toolkit.catmull_rom.performance"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.performance">Performance</a>
</h4>
<p>
The following performance numbers were generated for a call to the Catmull-Rom
interpolation method. The number that follows the slash is the number of points
passed to the interpolant.
</p>
<pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="number">2700</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x2</span><span class="special">)</span>
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">32</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x2</span><span class="special">)</span>
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">262</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x2</span><span class="special">)</span>
<span class="identifier">L3</span> <span class="identifier">Unified</span> <span class="number">3145</span><span class="identifier">K</span> <span class="special">(</span><span class="identifier">x1</span><span class="special">)</span>
<span class="special">---------------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span>
<span class="special">---------------------------------------------------------</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4</span> <span class="number">20</span> <span class="identifier">ns</span> <span class="number">20</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span> <span class="number">21</span> <span class="identifier">ns</span> <span class="number">21</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span> <span class="number">23</span> <span class="identifier">ns</span> <span class="number">23</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span> <span class="number">24</span> <span class="identifier">ns</span> <span class="number">24</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span> <span class="number">27</span> <span class="identifier">ns</span> <span class="number">27</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span> <span class="number">27</span> <span class="identifier">ns</span> <span class="number">27</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span> <span class="number">30</span> <span class="identifier">ns</span> <span class="number">30</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span> <span class="number">32</span> <span class="identifier">ns</span> <span class="number">31</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span> <span class="number">33</span> <span class="identifier">ns</span> <span class="number">33</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span> <span class="number">34</span> <span class="identifier">ns</span> <span class="number">34</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span> <span class="number">36</span> <span class="identifier">ns</span> <span class="number">36</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span> <span class="number">38</span> <span class="identifier">ns</span> <span class="number">38</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span> <span class="number">39</span> <span class="identifier">ns</span> <span class="number">39</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span> <span class="number">40</span> <span class="identifier">ns</span> <span class="number">40</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span> <span class="number">45</span> <span class="identifier">ns</span> <span class="number">44</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span> <span class="number">46</span> <span class="identifier">ns</span> <span class="number">46</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span> <span class="number">50</span> <span class="identifier">ns</span> <span class="number">50</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span> <span class="number">53</span> <span class="identifier">ns</span> <span class="number">52</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span> <span class="number">58</span> <span class="identifier">ns</span> <span class="number">57</span> <span class="identifier">ns</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span> <span class="number">2.97</span> <span class="identifier">lgN</span> <span class="number">2.97</span> <span class="identifier">lgN</span>
<span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span> <span class="number">19</span> <span class="special">%</span> <span class="number">19</span> <span class="special">%</span>
</pre>
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