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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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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>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.catmull_rom.h0"></a>
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<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>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span>
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<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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Point</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Point</span><span class="special">></span> <span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">catmull_rom</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">catmull_rom</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&&</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>
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<span class="identifier">catmull_rom</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">Point</span><span class="special">></span> <span class="identifier">l</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="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="special">(</span><span class="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">)</span> <span class="number">1</span><span class="special">/</span> <span class="special">(</span><span class="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">)</span> <span class="number">2</span><span class="special">);</span>
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<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>
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<span class="identifier">Real</span> <span class="identifier">max_parameter</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<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>
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<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>
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<span class="special">};</span>
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<span class="special">}}</span>
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</pre>
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<h4>
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<a name="math_toolkit.catmull_rom.h1"></a>
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<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>
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</h4>
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<p>
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Catmull-Rom splines are a family of interpolating curves which are commonly
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used in computer graphics and animation. Catmull-Rom splines enjoy the following
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properties:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Affine invariance: The interpolant commutes with affine transformations.
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</li>
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<li class="listitem">
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Local support of the basis functions: This gives stability and fast evaluation.
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</li>
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<li class="listitem">
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<span class="emphasis"><em>C</em></span><sup>2</sup>-smoothness
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</li>
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<li class="listitem">
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Interpolation of control points-this means the curve passes through the
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control points. Many curves (such as Bézier) are <span class="emphasis"><em>approximating</em></span>
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- they do not pass through their control points. This makes them more difficult
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to use than interpolating splines.
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</li>
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</ul></div>
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<p>
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The <code class="computeroutput"><span class="identifier">catmull_rom</span></code> class provided
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by Boost.Math creates a cubic Catmull-Rom spline from an array of points in
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any dimension. Since there are numerous ways to represent a point in <span class="emphasis"><em>n</em></span>-dimensional
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space, the class attempts to be flexible by templating on the point type. The
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requirements on the point type are discussing in more detail below, but roughly,
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it must have a dereference operator defined (e.g., <code class="computeroutput"><span class="identifier">p</span><span class="special">[</span><span class="number">0</span><span class="special">]</span></code>
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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>
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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
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requirements are met by <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
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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:
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</p>
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<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">>></span> <span class="identifier">points</span><span class="special">(</span><span class="number">4</span><span class="special">);</span>
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<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>
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<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>
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<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>
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<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>
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<span class="identifier">catmull_rom</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">>></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>
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<span class="comment">// Interpolate at s = 0.1:</span>
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<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>
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</pre>
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<p>
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The spline can be either open or <span class="emphasis"><em>closed</em></span>, closed meaning
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that there is some <span class="emphasis"><em>s > 0</em></span> such that <span class="emphasis"><em>P(s) =
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P(0)</em></span>. The default is open, but this can be easily changed:
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</p>
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<pre class="programlisting"><span class="comment">// closed = true</span>
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<span class="identifier">catmull_rom</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">>></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> <span class="keyword">true</span><span class="special">);</span>
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</pre>
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<p>
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In either case, evaluating the interpolator at <span class="emphasis"><em>s=0</em></span> returns
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the first point in the list.
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</p>
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<p>
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If the curve is open, then the first and last segments may have strange behavior.
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The traditional solution is to prepend a carefully selected control point to
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the data so that the first data segment (second interpolator segment) has reasonable
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tangent vectors, and simply ignore the first interpolator segment. A control
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point is appended to the data using similar criteria. However, we recommend
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not going through this effort until it proves to be necessary: For most use-cases,
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the curve is good enough without prepending and appending control points, and
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responsible selection of non-data control points is difficult.
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</p>
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<p>
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Inside <code class="computeroutput"><span class="identifier">catmull_rom</span></code>, the curve
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is represented as closed. This is because an open Catmull-Rom curve is <span class="emphasis"><em>implicitly
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closed</em></span>, but the closing point is the zero vector. There is no reason
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to suppose that the zero vector is a better closing point than the endpoint
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(or any other point, for that matter), so traditionally Catmull-Rom splines
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leave the segment between the first and second point undefined, as well as
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the segment between the second-to-last and last point. We find this property
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of the traditional implementation of Catmull-Rom splines annoying and confusing
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to the user. Hence internally, we close the curve so that the first and last
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segments are defined. Of course, this causes the <span class="emphasis"><em>tangent</em></span>
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vectors to the first and last points to be bizarre. This is a "pick your
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poison" design decision-either the curve cannot interpolate in its first
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and last segments, or the tangents along the first and last segments are meaningless.
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In the vast majority of cases, this will be no problem to the user. However,
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if it becomes a problem, then the user should add one extra point in a position
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they believe is reasonable and close the curve.
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</p>
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<p>
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Since the routine internally represents the curve as closed, a question arises:
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Why does the user have to specify if the curve is open or closed? The answer
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is that the parameterization is chosen by the routine, so it is of interest
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to the user to understand the values where a meaningful result is returned.
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</p>
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<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>
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</pre>
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<p>
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If you attempt to interpolate for <code class="computeroutput"><span class="identifier">s</span>
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<span class="special">></span> <span class="identifier">max_s</span></code>,
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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>
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<span class="special">=</span> <span class="identifier">p0</span></code>,
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where <code class="computeroutput"><span class="identifier">p0</span></code> is the first point
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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>
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<span class="special">=</span> <span class="identifier">pf</span></code>,
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where <code class="computeroutput"><span class="identifier">pf</span></code> is the final point
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on the curve.
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</p>
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<p>
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The Catmull-Rom curve admits an infinite number of parameterizations. The default
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parameterization of the <code class="computeroutput"><span class="identifier">catmull_rom</span></code>
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class is the so-called <span class="emphasis"><em>centripedal</em></span> parameterization. This
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parameterization has been shown to be the only parameterization that does not
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form cusps or self-intersections within segments. However, for advanced users,
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other parameterizations can be chosen using the <span class="emphasis"><em>alpha</em></span>
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parameter:
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</p>
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<pre class="programlisting"><span class="comment">// alpha = 1 is the "chordal" parameterization.</span>
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<span class="identifier">catmull_rom</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">3</span><span class="special">>></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> <span class="keyword">false</span><span class="special">,</span> <span class="number">1.0</span><span class="special">);</span>
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</pre>
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<p>
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The alpha parameter must always be in the range <code class="computeroutput"><span class="special">[</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">]</span></code>.
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</p>
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<p>
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Finally, the tangent vector to any point of the curve can be computed via
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</p>
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<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>
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<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>
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</pre>
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<p>
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Since the magnitude of the tangent vector is dependent on the parameterization,
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it is not meaningful (unless the user chooses the chordal parameterization
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<span class="emphasis"><em>alpha = 1</em></span> which parameterizes by Euclidean distance between
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points.) However, its direction is meaningful no matter the parameterization,
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so the user may wish to normalize this result.
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</p>
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<h4>
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<a name="math_toolkit.catmull_rom.h2"></a>
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<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>
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</h4>
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<h4>
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<a name="math_toolkit.catmull_rom.h3"></a>
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<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>
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</h4>
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<p>
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The following performance numbers were generated for a call to the Catmull-Rom
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interpolation method. The number that follows the slash is the number of points
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passed to the interpolant. We see that evaluation of the interpolant is 𝑶(<span class="emphasis"><em>log</em></span>(<span class="emphasis"><em>N</em></span>)).
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</p>
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<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>
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<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
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<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>
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<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>
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<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>
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<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>
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<span class="special">---------------------------------------------------------</span>
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<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span>
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<span class="special">---------------------------------------------------------</span>
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<span class="identifier">BM_CatmullRom</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">BM_CatmullRom</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">BM_CatmullRom</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">BM_CatmullRom</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
|
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<span class="identifier">BM_CatmullRom</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">></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"><</span><span class="keyword">double</span><span class="special">></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>
|
|
<h4>
|
|
<a name="math_toolkit.catmull_rom.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.catmull_rom.point_types"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.point_types">Point
|
|
types</a>
|
|
</h4>
|
|
<p>
|
|
We have already discussed that certain conditions on the <code class="computeroutput"><span class="identifier">Point</span></code>
|
|
type template argument must be obeyed. The following shows a custom point type
|
|
in 3D which can be used as a template argument to Catmull-Rom:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
|
|
<span class="keyword">class</span> <span class="identifier">mypoint3d</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">public</span><span class="special">:</span>
|
|
<span class="comment">// Must define a value_type:</span>
|
|
<span class="keyword">typedef</span> <span class="identifier">Real</span> <span class="identifier">value_type</span><span class="special">;</span>
|
|
|
|
<span class="comment">// Regular constructor--need not be of this form.</span>
|
|
<span class="identifier">mypoint3d</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">{</span><span class="identifier">m_vec</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">;</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">y</span><span class="special">;</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">z</span><span class="special">;</span> <span class="special">}</span>
|
|
|
|
<span class="comment">// Must define a default constructor:</span>
|
|
<span class="identifier">mypoint3d</span><span class="special">()</span> <span class="special">{}</span>
|
|
|
|
<span class="comment">// Must define array access:</span>
|
|
<span class="identifier">Real</span> <span class="keyword">operator</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="keyword">return</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="comment">// Must define array element assignment:</span>
|
|
<span class="identifier">Real</span><span class="special">&</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">return</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="keyword">private</span><span class="special">:</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">></span> <span class="identifier">m_vec</span><span class="special">;</span>
|
|
<span class="special">};</span>
|
|
|
|
|
|
<span class="comment">// Must define the free function "size()":</span>
|
|
<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
|
|
<span class="keyword">constexpr</span> <span class="identifier">size_t</span> <span class="identifier">size</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&</span> <span class="identifier">c</span><span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">return</span> <span class="number">3</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
</pre>
|
|
<p>
|
|
These conditions are satisfied by both <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code> and
|
|
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>, but it may nonetheless be useful
|
|
to define your own point class so that (say) you can define geometric distance
|
|
between them.
|
|
</p>
|
|
<h4>
|
|
<a name="math_toolkit.catmull_rom.h5"></a>
|
|
<span class="phrase"><a name="math_toolkit.catmull_rom.caveats"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.caveats">Caveats</a>
|
|
</h4>
|
|
<p>
|
|
The Catmull-Rom interpolator requires memory for three more points than is
|
|
provided by the user. This causes the class to call a <code class="computeroutput"><span class="identifier">resize</span><span class="special">()</span></code> on the input vector. If <code class="computeroutput"><span class="identifier">v</span><span class="special">.</span><span class="identifier">capacity</span><span class="special">()</span> <span class="special">>=</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span>
|
|
<span class="special">+</span> <span class="number">3</span></code>,
|
|
then no problems arise; there are no reallocs, and in practice this condition
|
|
is almost always satisfied. However, if <code class="computeroutput"><span class="identifier">v</span><span class="special">.</span><span class="identifier">capacity</span><span class="special">()</span> <span class="special"><</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span>
|
|
<span class="special">+</span> <span class="number">3</span></code>,
|
|
the <code class="computeroutput"><span class="identifier">realloc</span></code> causes a performance
|
|
penalty of roughly 20%.
|
|
</p>
|
|
<h4>
|
|
<a name="math_toolkit.catmull_rom.h6"></a>
|
|
<span class="phrase"><a name="math_toolkit.catmull_rom.generic_containers"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.generic_containers">Generic
|
|
Containers</a>
|
|
</h4>
|
|
<p>
|
|
The <code class="computeroutput"><span class="identifier">Point</span></code> type may be stored
|
|
in a different container than <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>.
|
|
For example, here is how to store the points in a Boost.uBLAS vector:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">p0</span><span class="special">(</span><span class="number">0.1</span><span class="special">,</span> <span class="number">0.2</span><span class="special">,</span> <span class="number">0.3</span><span class="special">);</span>
|
|
<span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">p1</span><span class="special">(</span><span class="number">0.2</span><span class="special">,</span> <span class="number">0.3</span><span class="special">,</span> <span class="number">0.4</span><span class="special">);</span>
|
|
<span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">p2</span><span class="special">(</span><span class="number">0.3</span><span class="special">,</span> <span class="number">0.4</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span>
|
|
<span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">p3</span><span class="special">(</span><span class="number">0.4</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.6</span><span class="special">);</span>
|
|
<span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">p4</span><span class="special">(</span><span class="number">0.5</span><span class="special">,</span> <span class="number">0.6</span><span class="special">,</span> <span class="number">0.7</span><span class="special">);</span>
|
|
<span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">p5</span><span class="special">(</span><span class="number">0.6</span><span class="special">,</span> <span class="number">0.7</span><span class="special">,</span> <span class="number">0.8</span><span class="special">);</span>
|
|
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>></span> <span class="identifier">u</span><span class="special">(</span><span class="number">6</span><span class="special">);</span>
|
|
<span class="identifier">u</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p0</span><span class="special">;</span>
|
|
<span class="identifier">u</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p1</span><span class="special">;</span>
|
|
<span class="identifier">u</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p2</span><span class="special">;</span>
|
|
<span class="identifier">u</span><span class="special">[</span><span class="number">3</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p3</span><span class="special">;</span>
|
|
<span class="identifier">u</span><span class="special">[</span><span class="number">4</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p4</span><span class="special">;</span>
|
|
<span class="identifier">u</span><span class="special">[</span><span class="number">5</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p5</span><span class="special">;</span>
|
|
|
|
<span class="comment">// Tests initializer_list:</span>
|
|
<span class="identifier">catmull_rom</span><span class="special"><</span><span class="identifier">mypoint3d</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>,</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">u</span><span class="special">)></span> <span class="identifier">cat</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">u</span><span class="special">));</span>
|
|
</pre>
|
|
<h4>
|
|
<a name="math_toolkit.catmull_rom.h7"></a>
|
|
<span class="phrase"><a name="math_toolkit.catmull_rom.references"></a></span><a class="link" href="catmull_rom.html#math_toolkit.catmull_rom.references">References</a>
|
|
</h4>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
Cem Yuksel, Scott Schaefer, and John Keyser, <span class="emphasis"><em>Parameterization
|
|
and applications of Catmull–Rom curves</em></span>, Computer-Aided Design
|
|
43 (2011) 747–755.
|
|
</li>
|
|
<li class="listitem">
|
|
Phillip J. Barry and Ronald N. Goldman, <span class="emphasis"><em>A Recursive Evaluation
|
|
Algorithm for a Class of Catmull-Rom Splines</em></span>, Computer Graphics,
|
|
Volume 22, Number 4, August 1988
|
|
</li>
|
|
</ul></div>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
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