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    25. 

  25. <div class="section">
    26. 

  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
    27. 

  27. <a name="math_toolkit.catmull_rom"></a><a class="link" href="catmull_rom.html" title="Catmull-Rom Splines">Catmull-Rom Splines</a>
    28. 

  28. </h2></div></div></div>
    29. 

  29. <h4>
    30. 

  30. <a name="math_toolkit.catmull_rom.h0"></a>
    31. 

  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. 

  32.     </h4>
    33. 

  33. <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">catmull_rom</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    34. 

  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. 

  36. 

  37.     <span class="keyword">template</span><span class="special">&lt;</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">&lt;</span><span class="identifier">Point</span><span class="special">&gt;</span> <span class="special">&gt;</span>
    38. 

  38.     <span class="keyword">class</span> <span class="identifier">catmull_rom</span>
    39. 

  39.     <span class="special">{</span>
    40. 

  40.     <span class="keyword">public</span><span class="special">:</span>
    41. 

  41. 

  42.         <span class="identifier">catmull_rom</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">points</span><span class="special">,</span> <span class="keyword">bool</span> <span class="identifier">closed</span> <span class="special">=</span> <span class="keyword">false</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">1</span><span class="special">/</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">2</span><span class="special">)</span>
    43. 

  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">&lt;</span><span class="identifier">Point</span><span class="special">&gt;</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. 

  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. 

  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. 

  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. 

  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. 

  53.     <span class="special">};</span>
    54. 

  54. 

  55. <span class="special">}}</span>
    56. 

  56. </pre>
    57. 

  57. <h4>
    58. 

  58. <a name="math_toolkit.catmull_rom.h1"></a>
    59. 

  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. 

  60.     </h4>
    61. 

  61. <p>
    62. 

  62.       Catmull-Rom splines are a family of interpolating curves which are commonly
    63. 

  63.       used in computer graphics and animation. Catmull-Rom splines enjoy the following
    64. 

  64.       properties:
    65. 

  65.     </p>
    66. 

  66. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    67. 

  67. <li class="listitem">
    68. 

  68.           Affine invariance: The interpolant commutes with affine transformations.
    69. 

  69.         </li>
    70. 

  70. <li class="listitem">
    71. 

  71.           Local support of the basis functions: This gives stability and fast evaluation.
    72. 

  72.         </li>
    73. 

  73. <li class="listitem">
    74. 

  74.           <span class="emphasis"><em>C</em></span><sup>2</sup>-smoothness
    75. 

  75.         </li>
    76. 

  76. <li class="listitem">
    77. 

  77.           Interpolation of control points-this means the curve passes through the
    78. 

  78.           control points. Many curves (such as B&#233;zier) are <span class="emphasis"><em>approximating</em></span>
    79. 

  79.           - they do not pass through their control points. This makes them more difficult
    80. 

  80.           to use than interpolating splines.
    81. 

  81.         </li>
    82. 

  82. </ul></div>
    83. 

  83. <p>
    84. 

  84.       The <code class="computeroutput"><span class="identifier">catmull_rom</span></code> class provided
    85. 

  85.       by Boost.Math creates a cubic Catmull-Rom spline from an array of points in
    86. 

  86.       any dimension. Since there are numerous ways to represent a point in <span class="emphasis"><em>n</em></span>-dimensional
    87. 

  87.       space, the class attempts to be flexible by templating on the point type. The
    88. 

  88.       requirements on the point type are discussing in more detail below, but roughly,
    89. 

  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. 

  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. 

  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. 

  92.       requirements are met by <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>
    93. 

  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. 

  94.     </p>
    95. 

  95. <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">points</span><span class="special">(</span><span class="number">4</span><span class="special">);</span>
    96. 

  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. 

  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. 

  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. 

  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. 

  100. <span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">points</span><span class="special">));</span>
    101. 

  101. <span class="comment">// Interpolate at s = 0.1:</span>
    102. 

  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. 

  103. </pre>
    104. 

  104. <p>
    105. 

  105.       The spline can be either open or <span class="emphasis"><em>closed</em></span>, closed meaning
    106. 

  106.       that there is some <span class="emphasis"><em>s &gt; 0</em></span> such that <span class="emphasis"><em>P(s) =
    107. 

  107.       P(0)</em></span>. The default is open, but this can be easily changed:
    108. 

  108.     </p>
    109. 

  109. <pre class="programlisting"><span class="comment">// closed = true</span>
    110. 

  110. <span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">points</span><span class="special">),</span> <span class="keyword">true</span><span class="special">);</span>
    111. 

  111. </pre>
    112. 

  112. <p>
    113. 

  113.       In either case, evaluating the interpolator at <span class="emphasis"><em>s=0</em></span> returns
    114. 

  114.       the first point in the list.
    115. 

  115.     </p>
    116. 

  116. <p>
    117. 

  117.       If the curve is open, then the first and last segments may have strange behavior.
    118. 

  118.       The traditional solution is to prepend a carefully selected control point to
    119. 

  119.       the data so that the first data segment (second interpolator segment) has reasonable
    120. 

  120.       tangent vectors, and simply ignore the first interpolator segment. A control
    121. 

  121.       point is appended to the data using similar criteria. However, we recommend
    122. 

  122.       not going through this effort until it proves to be necessary: For most use-cases,
    123. 

  123.       the curve is good enough without prepending and appending control points, and
    124. 

  124.       responsible selection of non-data control points is difficult.
    125. 

  125.     </p>
    126. 

  126. <p>
    127. 

  127.       Inside <code class="computeroutput"><span class="identifier">catmull_rom</span></code>, the curve
    128. 

  128.       is represented as closed. This is because an open Catmull-Rom curve is <span class="emphasis"><em>implicitly
    129. 

  129.       closed</em></span>, but the closing point is the zero vector. There is no reason
    130. 

  130.       to suppose that the zero vector is a better closing point than the endpoint
    131. 

  131.       (or any other point, for that matter), so traditionally Catmull-Rom splines
    132. 

  132.       leave the segment between the first and second point undefined, as well as
    133. 

  133.       the segment between the second-to-last and last point. We find this property
    134. 

  134.       of the traditional implementation of Catmull-Rom splines annoying and confusing
    135. 

  135.       to the user. Hence internally, we close the curve so that the first and last
    136. 

  136.       segments are defined. Of course, this causes the <span class="emphasis"><em>tangent</em></span>
    137. 

  137.       vectors to the first and last points to be bizarre. This is a "pick your
    138. 

  138.       poison" design decision-either the curve cannot interpolate in its first
    139. 

  139.       and last segments, or the tangents along the first and last segments are meaningless.
    140. 

  140.       In the vast majority of cases, this will be no problem to the user. However,
    141. 

  141.       if it becomes a problem, then the user should add one extra point in a position
    142. 

  142.       they believe is reasonable and close the curve.
    143. 

  143.     </p>
    144. 

  144. <p>
    145. 

  145.       Since the routine internally represents the curve as closed, a question arises:
    146. 

  146.       Why does the user have to specify if the curve is open or closed? The answer
    147. 

  147.       is that the parameterization is chosen by the routine, so it is of interest
    148. 

  148.       to the user to understand the values where a meaningful result is returned.
    149. 

  149.     </p>
    150. 

  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. 

  151. </pre>
    152. 

  152. <p>
    153. 

  153.       If you attempt to interpolate for <code class="computeroutput"><span class="identifier">s</span>
    154. 

  154.       <span class="special">&gt;</span> <span class="identifier">max_s</span></code>,
    155. 

  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. 

  156.       <span class="special">=</span> <span class="identifier">p0</span></code>,
    157. 

  157.       where <code class="computeroutput"><span class="identifier">p0</span></code> is the first point
    158. 

  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. 

  159.       <span class="special">=</span> <span class="identifier">pf</span></code>,
    160. 

  160.       where <code class="computeroutput"><span class="identifier">pf</span></code> is the final point
    161. 

  161.       on the curve.
    162. 

  162.     </p>
    163. 

  163. <p>
    164. 

  164.       The Catmull-Rom curve admits an infinite number of parameterizations. The default
    165. 

  165.       parameterization of the <code class="computeroutput"><span class="identifier">catmull_rom</span></code>
    166. 

  166.       class is the so-called <span class="emphasis"><em>centripedal</em></span> parameterization. This
    167. 

  167.       parameterization has been shown to be the only parameterization that does not
    168. 

  168.       form cusps or self-intersections within segments. However, for advanced users,
    169. 

  169.       other parameterizations can be chosen using the <span class="emphasis"><em>alpha</em></span>
    170. 

  170.       parameter:
    171. 

  171.     </p>
    172. 

  172. <pre class="programlisting"><span class="comment">// alpha = 1 is the "chordal" parameterization.</span>
    173. 

  173. <span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;&gt;</span> <span class="identifier">cr</span><span class="special">(</span><span class="identifier">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. 

  174. </pre>
    175. 

  175. <p>
    176. 

  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. 

  177.     </p>
    178. 

  178. <p>
    179. 

  179.       Finally, the tangent vector to any point of the curve can be computed via
    180. 

  180.     </p>
    181. 

  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. 

  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. 

  183. </pre>
    184. 

  184. <p>
    185. 

  185.       Since the magnitude of the tangent vector is dependent on the parameterization,
    186. 

  186.       it is not meaningful (unless the user chooses the chordal parameterization
    187. 

  187.       <span class="emphasis"><em>alpha = 1</em></span> which parameterizes by Euclidean distance between
    188. 

  188.       points.) However, its direction is meaningful no matter the parameterization,
    189. 

  189.       so the user may wish to normalize this result.
    190. 

  190.     </p>
    191. 

  191. <h4>
    192. 

  192. <a name="math_toolkit.catmull_rom.h2"></a>
    193. 

  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. 

  194.     </h4>
    195. 

  195. <h4>
    196. 

  196. <a name="math_toolkit.catmull_rom.h3"></a>
    197. 

  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. 

  198.     </h4>
    199. 

  199. <p>
    200. 

  200.       The following performance numbers were generated for a call to the Catmull-Rom
    201. 

  201.       interpolation method. The number that follows the slash is the number of points
    202. 

  202.       passed to the interpolant. We see that evaluation of the interpolant is &#119926;(<span class="emphasis"><em>log</em></span>(<span class="emphasis"><em>N</em></span>)).
    203. 

  203.     </p>
    204. 

  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. 

  205. <span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
    206. 

  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. 

  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. 

  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. 

  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. 

  210. <span class="special">---------------------------------------------------------</span>
    211. 

  211. <span class="identifier">Benchmark</span>                              <span class="identifier">Time</span>           <span class="identifier">CPU</span>
    212. 

  212. <span class="special">---------------------------------------------------------</span>
    213. 

  213. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4</span>               <span class="number">20</span> <span class="identifier">ns</span>         <span class="number">20</span> <span class="identifier">ns</span>
    214. 

  214. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">21</span> <span class="identifier">ns</span>         <span class="number">21</span> <span class="identifier">ns</span>
    215. 

  215. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>              <span class="number">23</span> <span class="identifier">ns</span>         <span class="number">23</span> <span class="identifier">ns</span>
    216. 

  216. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>              <span class="number">24</span> <span class="identifier">ns</span>         <span class="number">24</span> <span class="identifier">ns</span>
    217. 

  217. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>              <span class="number">27</span> <span class="identifier">ns</span>         <span class="number">27</span> <span class="identifier">ns</span>
    218. 

  218. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>             <span class="number">27</span> <span class="identifier">ns</span>         <span class="number">27</span> <span class="identifier">ns</span>
    219. 

  219. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>             <span class="number">30</span> <span class="identifier">ns</span>         <span class="number">30</span> <span class="identifier">ns</span>
    220. 

  220. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>             <span class="number">32</span> <span class="identifier">ns</span>         <span class="number">31</span> <span class="identifier">ns</span>
    221. 

  221. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>            <span class="number">33</span> <span class="identifier">ns</span>         <span class="number">33</span> <span class="identifier">ns</span>
    222. 

  222. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>            <span class="number">34</span> <span class="identifier">ns</span>         <span class="number">34</span> <span class="identifier">ns</span>
    223. 

  223. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>            <span class="number">36</span> <span class="identifier">ns</span>         <span class="number">36</span> <span class="identifier">ns</span>
    224. 

  224. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>            <span class="number">38</span> <span class="identifier">ns</span>         <span class="number">38</span> <span class="identifier">ns</span>
    225. 

  225. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>           <span class="number">39</span> <span class="identifier">ns</span>         <span class="number">39</span> <span class="identifier">ns</span>
    226. 

  226. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>           <span class="number">40</span> <span class="identifier">ns</span>         <span class="number">40</span> <span class="identifier">ns</span>
    227. 

  227. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>           <span class="number">45</span> <span class="identifier">ns</span>         <span class="number">44</span> <span class="identifier">ns</span>
    228. 

  228. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>          <span class="number">46</span> <span class="identifier">ns</span>         <span class="number">46</span> <span class="identifier">ns</span>
    229. 

  229. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>          <span class="number">50</span> <span class="identifier">ns</span>         <span class="number">50</span> <span class="identifier">ns</span>
    230. 

  230. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>          <span class="number">53</span> <span class="identifier">ns</span>         <span class="number">52</span> <span class="identifier">ns</span>
    231. 

  231. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>         <span class="number">58</span> <span class="identifier">ns</span>         <span class="number">57</span> <span class="identifier">ns</span>
    232. 

  232. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>          <span class="number">2.97</span> <span class="identifier">lgN</span>       <span class="number">2.97</span> <span class="identifier">lgN</span>
    233. 

  233. <span class="identifier">BM_CatmullRom</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span>             <span class="number">19</span> <span class="special">%</span>         <span class="number">19</span> <span class="special">%</span>
    234. 

  234. </pre>
    235. 

  235. <h4>
    236. 

  236. <a name="math_toolkit.catmull_rom.h4"></a>
    237. 

  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. 

  238.       types</a>
    239. 

  239.     </h4>
    240. 

  240. <p>
    241. 

  241.       We have already discussed that certain conditions on the <code class="computeroutput"><span class="identifier">Point</span></code>
    242. 

  242.       type template argument must be obeyed. The following shows a custom point type
    243. 

  243.       in 3D which can be used as a template argument to Catmull-Rom:
    244. 

  244.     </p>
    245. 

  245. <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
    246. 

  246. <span class="keyword">class</span> <span class="identifier">mypoint3d</span>
    247. 

  247. <span class="special">{</span>
    248. 

  248. <span class="keyword">public</span><span class="special">:</span>
    249. 

  249.     <span class="comment">// Must define a value_type:</span>
    250. 

  250.     <span class="keyword">typedef</span> <span class="identifier">Real</span> <span class="identifier">value_type</span><span class="special">;</span>
    251. 

  251. 

  252.     <span class="comment">// Regular constructor--need not be of this form.</span>
    253. 

  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. 

  254. 

  255.     <span class="comment">// Must define a default constructor:</span>
    256. 

  256.     <span class="identifier">mypoint3d</span><span class="special">()</span> <span class="special">{}</span>
    257. 

  257. 

  258.     <span class="comment">// Must define array access:</span>
    259. 

  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. 

  260.     <span class="special">{</span>
    261. 

  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. 

  262.     <span class="special">}</span>
    263. 

  263. 

  264.     <span class="comment">// Must define array element assignment:</span>
    265. 

  265.     <span class="identifier">Real</span><span class="special">&amp;</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. 

  266.     <span class="special">{</span>
    267. 

  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. 

  268.     <span class="special">}</span>
    269. 

  269. 

  270. <span class="keyword">private</span><span class="special">:</span>
    271. 

  271.     <span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">&gt;</span> <span class="identifier">m_vec</span><span class="special">;</span>
    272. 

  272. <span class="special">};</span>
    273. 

  273. 

  274. 

  275. <span class="comment">// Must define the free function "size()":</span>
    276. 

  276. <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
    277. 

  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">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;</span> <span class="identifier">c</span><span class="special">)</span>
    278. 

  278. <span class="special">{</span>
    279. 

  279.     <span class="keyword">return</span> <span class="number">3</span><span class="special">;</span>
    280. 

  280. <span class="special">}</span>
    281. 

  281. </pre>
    282. 

  282. <p>
    283. 

  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. 

  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. 

  285.       to define your own point class so that (say) you can define geometric distance
    286. 

  286.       between them.
    287. 

  287.     </p>
    288. 

  288. <h4>
    289. 

  289. <a name="math_toolkit.catmull_rom.h5"></a>
    290. 

  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. 

  291.     </h4>
    292. 

  292. <p>
    293. 

  293.       The Catmull-Rom interpolator requires memory for three more points than is
    294. 

  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">&gt;=</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span>
    295. 

  295.       <span class="special">+</span> <span class="number">3</span></code>,
    296. 

  296.       then no problems arise; there are no reallocs, and in practice this condition
    297. 

  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">&lt;</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span>
    298. 

  298.       <span class="special">+</span> <span class="number">3</span></code>,
    299. 

  299.       the <code class="computeroutput"><span class="identifier">realloc</span></code> causes a performance
    300. 

  300.       penalty of roughly 20%.
    301. 

  301.     </p>
    302. 

  302. <h4>
    303. 

  303. <a name="math_toolkit.catmull_rom.h6"></a>
    304. 

  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. 

  305.       Containers</a>
    306. 

  306.     </h4>
    307. 

  307. <p>
    308. 

  308.       The <code class="computeroutput"><span class="identifier">Point</span></code> type may be stored
    309. 

  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. 

  310.       For example, here is how to store the points in a Boost.uBLAS vector:
    311. 

  311.     </p>
    312. 

  312. <pre class="programlisting"><span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</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. 

  313. <span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</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. 

  314. <span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</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. 

  315. <span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</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. 

  316. <span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</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. 

  317. <span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;</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. 

  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">&lt;</span><span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&gt;</span> <span class="identifier">u</span><span class="special">(</span><span class="number">6</span><span class="special">);</span>
    320. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  326. 

  327. <span class="comment">// Tests initializer_list:</span>
    328. 

  328. <span class="identifier">catmull_rom</span><span class="special">&lt;</span><span class="identifier">mypoint3d</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;,</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">u</span><span class="special">)&gt;</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. 

  329. </pre>
    330. 

  330. <h4>
    331. 

  331. <a name="math_toolkit.catmull_rom.h7"></a>
    332. 

  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. 

  333.     </h4>
    334. 

  334. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    335. 

  335. <li class="listitem">
    336. 

  336.           Cem Yuksel, Scott Schaefer, and John Keyser, <span class="emphasis"><em>Parameterization
    337. 

  337.           and applications of Catmull&#8211;Rom curves</em></span>, Computer-Aided Design
    338. 

  338.           43 (2011) 747&#8211;755.
    339. 

  339.         </li>
    340. 

  340. <li class="listitem">
    341. 

  341.           Phillip J. Barry and Ronald N. Goldman, <span class="emphasis"><em>A Recursive Evaluation
    342. 

  342.           Algorithm for a Class of Catmull-Rom Splines</em></span>, Computer Graphics,
    343. 

  343.           Volume 22, Number 4, August 1988
    344. 

  344.         </li>
    345. 

  345. </ul></div>
    346. 

  346. </div>
    347. 

  347. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    348. 

  348. <td align="left"></td>
    349. 

  349. <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
    350. 

  350.       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
    351. 

  351.       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
    352. 

  352.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
    353. 

  353.       Daryle Walker and Xiaogang Zhang<p>
    354. 

  354.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    355. 

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