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Interpolation"> 9<link rel="prev" href="vector_barycentric.html" title="Vector-valued Barycentric Rational Interpolation"> 10<link rel="next" href="cardinal_trigonometric.html" title="Cardinal Trigonometric interpolation"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="vector_barycentric.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../interpolation.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cardinal_trigonometric.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.catmull_rom"></a><a class="link" href="catmull_rom.html" title="Catmull-Rom Splines">Catmull-Rom Splines</a> 28</h2></div></div></div> 29<h4> 30<a name="math_toolkit.catmull_rom.h0"></a> 31 <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> 32 </h4> 33<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> 34 35<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> 36 37 <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> 38 <span class="keyword">class</span> <span class="identifier">catmull_rom</span> 39 <span class="special">{</span> 40 <span class="keyword">public</span><span class="special">:</span> 41 42 <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> 43 44 <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> 45 46 <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> 47 48 <span class="identifier">Real</span> <span class="identifier">max_parameter</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> 49 50 <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> 51 52 <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> 53 <span class="special">};</span> 54 55<span class="special">}}</span> 56</pre> 57<h4> 58<a name="math_toolkit.catmull_rom.h1"></a> 59 <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> 60 </h4> 61<p> 62 Catmull-Rom splines are a family of interpolating curves which are commonly 63 used in computer graphics and animation. Catmull-Rom splines enjoy the following 64 properties: 65 </p> 66<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 67<li class="listitem"> 68 Affine invariance: The interpolant commutes with affine transformations. 69 </li> 70<li class="listitem"> 71 Local support of the basis functions: This gives stability and fast evaluation. 72 </li> 73<li class="listitem"> 74 <span class="emphasis"><em>C</em></span><sup>2</sup>-smoothness 75 </li> 76<li class="listitem"> 77 Interpolation of control points-this means the curve passes through the 78 control points. Many curves (such as Bézier) are <span class="emphasis"><em>approximating</em></span> 79 - they do not pass through their control points. This makes them more difficult 80 to use than interpolating splines. 81 </li> 82</ul></div> 83<p> 84 The <code class="computeroutput"><span class="identifier">catmull_rom</span></code> class provided 85 by Boost.Math creates a cubic Catmull-Rom spline from an array of points in 86 any dimension. Since there are numerous ways to represent a point in <span class="emphasis"><em>n</em></span>-dimensional 87 space, the class attempts to be flexible by templating on the point type. The 88 requirements on the point type are discussing in more detail below, but roughly, 89 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> 90 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> 91 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 92 requirements are met by <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code> 93 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: 94 </p> 95<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> 96<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> 97<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> 98<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> 99<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> 100<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> 101<span class="comment">// Interpolate at s = 0.1:</span> 102<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> 103</pre> 104<p> 105 The spline can be either open or <span class="emphasis"><em>closed</em></span>, closed meaning 106 that there is some <span class="emphasis"><em>s > 0</em></span> such that <span class="emphasis"><em>P(s) = 107 P(0)</em></span>. The default is open, but this can be easily changed: 108 </p> 109<pre class="programlisting"><span class="comment">// closed = true</span> 110<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> 111</pre> 112<p> 113 In either case, evaluating the interpolator at <span class="emphasis"><em>s=0</em></span> returns 114 the first point in the list. 115 </p> 116<p> 117 If the curve is open, then the first and last segments may have strange behavior. 118 The traditional solution is to prepend a carefully selected control point to 119 the data so that the first data segment (second interpolator segment) has reasonable 120 tangent vectors, and simply ignore the first interpolator segment. A control 121 point is appended to the data using similar criteria. However, we recommend 122 not going through this effort until it proves to be necessary: For most use-cases, 123 the curve is good enough without prepending and appending control points, and 124 responsible selection of non-data control points is difficult. 125 </p> 126<p> 127 Inside <code class="computeroutput"><span class="identifier">catmull_rom</span></code>, the curve 128 is represented as closed. This is because an open Catmull-Rom curve is <span class="emphasis"><em>implicitly 129 closed</em></span>, but the closing point is the zero vector. There is no reason 130 to suppose that the zero vector is a better closing point than the endpoint 131 (or any other point, for that matter), so traditionally Catmull-Rom splines 132 leave the segment between the first and second point undefined, as well as 133 the segment between the second-to-last and last point. We find this property 134 of the traditional implementation of Catmull-Rom splines annoying and confusing 135 to the user. Hence internally, we close the curve so that the first and last 136 segments are defined. Of course, this causes the <span class="emphasis"><em>tangent</em></span> 137 vectors to the first and last points to be bizarre. This is a "pick your 138 poison" design decision-either the curve cannot interpolate in its first 139 and last segments, or the tangents along the first and last segments are meaningless. 140 In the vast majority of cases, this will be no problem to the user. However, 141 if it becomes a problem, then the user should add one extra point in a position 142 they believe is reasonable and close the curve. 143 </p> 144<p> 145 Since the routine internally represents the curve as closed, a question arises: 146 Why does the user have to specify if the curve is open or closed? The answer 147 is that the parameterization is chosen by the routine, so it is of interest 148 to the user to understand the values where a meaningful result is returned. 149 </p> 150<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> 151</pre> 152<p> 153 If you attempt to interpolate for <code class="computeroutput"><span class="identifier">s</span> 154 <span class="special">></span> <span class="identifier">max_s</span></code>, 155 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> 156 <span class="special">=</span> <span class="identifier">p0</span></code>, 157 where <code class="computeroutput"><span class="identifier">p0</span></code> is the first point 158 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> 159 <span class="special">=</span> <span class="identifier">pf</span></code>, 160 where <code class="computeroutput"><span class="identifier">pf</span></code> is the final point 161 on the curve. 162 </p> 163<p> 164 The Catmull-Rom curve admits an infinite number of parameterizations. The default 165 parameterization of the <code class="computeroutput"><span class="identifier">catmull_rom</span></code> 166 class is the so-called <span class="emphasis"><em>centripedal</em></span> parameterization. This 167 parameterization has been shown to be the only parameterization that does not 168 form cusps or self-intersections within segments. However, for advanced users, 169 other parameterizations can be chosen using the <span class="emphasis"><em>alpha</em></span> 170 parameter: 171 </p> 172<pre class="programlisting"><span class="comment">// alpha = 1 is the "chordal" parameterization.</span> 173<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> 174</pre> 175<p> 176 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>. 177 </p> 178<p> 179 Finally, the tangent vector to any point of the curve can be computed via 180 </p> 181<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> 182<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> 183</pre> 184<p> 185 Since the magnitude of the tangent vector is dependent on the parameterization, 186 it is not meaningful (unless the user chooses the chordal parameterization 187 <span class="emphasis"><em>alpha = 1</em></span> which parameterizes by Euclidean distance between 188 points.) However, its direction is meaningful no matter the parameterization, 189 so the user may wish to normalize this result. 190 </p> 191<h4> 192<a name="math_toolkit.catmull_rom.h2"></a> 193 <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> 194 </h4> 195<h4> 196<a name="math_toolkit.catmull_rom.h3"></a> 197 <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> 198 </h4> 199<p> 200 The following performance numbers were generated for a call to the Catmull-Rom 201 interpolation method. The number that follows the slash is the number of points 202 passed to the interpolant. We see that evaluation of the interpolant is ��(<span class="emphasis"><em>log</em></span>(<span class="emphasis"><em>N</em></span>)). 203 </p> 204<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> 205<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span> 206 <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> 207 <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> 208 <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> 209 <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> 210<span class="special">---------------------------------------------------------</span> 211<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span> 212<span class="special">---------------------------------------------------------</span> 213<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> 214<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> 215<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> 216<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> 217<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> 218<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> 219<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> 220<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> 221<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> 222<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> 223<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> 224<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> 225<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> 226<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> 227<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> 228<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> 229<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> 230<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> 231<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> 232<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> 233<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> 234</pre> 235<h4> 236<a name="math_toolkit.catmull_rom.h4"></a> 237 <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 238 types</a> 239 </h4> 240<p> 241 We have already discussed that certain conditions on the <code class="computeroutput"><span class="identifier">Point</span></code> 242 type template argument must be obeyed. The following shows a custom point type 243 in 3D which can be used as a template argument to Catmull-Rom: 244 </p> 245<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> 246<span class="keyword">class</span> <span class="identifier">mypoint3d</span> 247<span class="special">{</span> 248<span class="keyword">public</span><span class="special">:</span> 249 <span class="comment">// Must define a value_type:</span> 250 <span class="keyword">typedef</span> <span class="identifier">Real</span> <span class="identifier">value_type</span><span class="special">;</span> 251 252 <span class="comment">// Regular constructor--need not be of this form.</span> 253 <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> 254 255 <span class="comment">// Must define a default constructor:</span> 256 <span class="identifier">mypoint3d</span><span class="special">()</span> <span class="special">{}</span> 257 258 <span class="comment">// Must define array access:</span> 259 <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> 260 <span class="special">{</span> 261 <span class="keyword">return</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span> 262 <span class="special">}</span> 263 264 <span class="comment">// Must define array element assignment:</span> 265 <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> 266 <span class="special">{</span> 267 <span class="keyword">return</span> <span class="identifier">m_vec</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span> 268 <span class="special">}</span> 269 270<span class="keyword">private</span><span class="special">:</span> 271 <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> 272<span class="special">};</span> 273 274 275<span class="comment">// Must define the free function "size()":</span> 276<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span> 277<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> 278<span class="special">{</span> 279 <span class="keyword">return</span> <span class="number">3</span><span class="special">;</span> 280<span class="special">}</span> 281</pre> 282<p> 283 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 284 <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>, but it may nonetheless be useful 285 to define your own point class so that (say) you can define geometric distance 286 between them. 287 </p> 288<h4> 289<a name="math_toolkit.catmull_rom.h5"></a> 290 <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> 291 </h4> 292<p> 293 The Catmull-Rom interpolator requires memory for three more points than is 294 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> 295 <span class="special">+</span> <span class="number">3</span></code>, 296 then no problems arise; there are no reallocs, and in practice this condition 297 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> 298 <span class="special">+</span> <span class="number">3</span></code>, 299 the <code class="computeroutput"><span class="identifier">realloc</span></code> causes a performance 300 penalty of roughly 20%. 301 </p> 302<h4> 303<a name="math_toolkit.catmull_rom.h6"></a> 304 <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 305 Containers</a> 306 </h4> 307<p> 308 The <code class="computeroutput"><span class="identifier">Point</span></code> type may be stored 309 in a different container than <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>. 310 For example, here is how to store the points in a Boost.uBLAS vector: 311 </p> 312<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> 313<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> 314<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> 315<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> 316<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> 317<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> 318 319<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> 320<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> 321<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> 322<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> 323<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> 324<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> 325<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> 326 327<span class="comment">// Tests initializer_list:</span> 328<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> 329</pre> 330<h4> 331<a name="math_toolkit.catmull_rom.h7"></a> 332 <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> 333 </h4> 334<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 335<li class="listitem"> 336 Cem Yuksel, Scott Schaefer, and John Keyser, <span class="emphasis"><em>Parameterization 337 and applications of Catmull–Rom curves</em></span>, Computer-Aided Design 338 43 (2011) 747–755. 339 </li> 340<li class="listitem"> 341 Phillip J. Barry and Ronald N. Goldman, <span class="emphasis"><em>A Recursive Evaluation 342 Algorithm for a Class of Catmull-Rom Splines</em></span>, Computer Graphics, 343 Volume 22, Number 4, August 1988 344 </li> 345</ul></div> 346</div> 347<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 348<td align="left"></td> 349<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 350 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 351 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 352 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 353 Daryle Walker and Xiaogang Zhang<p> 354 Distributed under the Boost Software License, Version 1.0. (See accompanying 355 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>) 356 </p> 357</div></td> 358</tr></table> 359<hr> 360<div class="spirit-nav"> 361<a accesskey="p" href="vector_barycentric.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../interpolation.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cardinal_trigonometric.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 362</div> 363</body> 364</html> 365
