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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.bezier_polynomial"></a><a class="link" href="bezier_polynomial.html" title="Bezier Polynomials">Bezier Polynomials</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.bezier_polynomial.h0"></a>
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<span class="phrase"><a name="math_toolkit.bezier_polynomial.synopsis"></a></span><a class="link" href="bezier_polynomial.html#math_toolkit.bezier_polynomial.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">bezier_polynomials</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="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span> <span class="special">{</span>
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<span class="keyword">template</span><span class="special"><</span><span class="identifier">RandomAccessContainer</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">bezier_polynomial</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="keyword">using</span> <span class="identifier">Point</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="identifier">Real</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>
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<span class="keyword">using</span> <span class="identifier">Z</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">size_type</span><span class="special">;</span>
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<span class="identifier">bezier_polynomial</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&&</span> <span class="identifier">control_points</span><span class="special">);</span>
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<span class="keyword">inline</span> <span class="identifier">Point</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">inline</span> <span class="identifier">Point</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">void</span> <span class="identifier">edit_control_point</span><span class="special">(</span><span class="identifier">Point</span> <span class="identifier">cont</span> <span class="special">&</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">Z</span> <span class="identifier">index</span><span class="special">);</span>
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<span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">control_points</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">friend</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="keyword">operator</span><span class="special"><<(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="identifier">out</span><span class="special">,</span> <span class="identifier">bezier_polynomial</span><span class="special"><</span><span class="identifier">RandomAccessContainer</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">bp</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.bezier_polynomial.h1"></a>
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<span class="phrase"><a name="math_toolkit.bezier_polynomial.description"></a></span><a class="link" href="bezier_polynomial.html#math_toolkit.bezier_polynomial.description">Description</a>
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</h4>
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<p>
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Bézier polynomials are curves smooth curves which approximate a set of control
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points. They are commonly used in computer-aided geometric design. A basic
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usage is demonstrated below:
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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="keyword">double</span><span class="special">,</span> <span class="number">3</span><span class="special">>></span> <span class="identifier">control_points</span><span class="special">(</span><span class="number">4</span><span class="special">);</span>
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<span class="identifier">control_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">control_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">control_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">control_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="keyword">auto</span> <span class="identifier">bp</span> <span class="special">=</span> <span class="identifier">bezier_polynomial</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">control_points</span><span class="special">));</span>
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<span class="comment">// Interpolate at t = 0.1:</span>
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<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">point</span> <span class="special">=</span> <span class="identifier">bp</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 support of the interpolant is [0,1], and an error message will be written
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if attempting to evaluate the polynomial outside of these bounds. At least
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two points must be passed; creating a polynomial of degree 1.
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</p>
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<p>
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Control points may be modified via <code class="computeroutput"><span class="identifier">edit_control_point</span></code>,
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for example:
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</p>
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<pre class="programlisting"><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">endpoint</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">1</span><span class="special">};</span>
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<span class="identifier">bp</span><span class="special">.</span><span class="identifier">edit_control_point</span><span class="special">(</span><span class="identifier">endpoint</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span>
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</pre>
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<p>
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This replaces the last control point with <code class="computeroutput"><span class="identifier">endpoint</span></code>.
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</p>
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<p>
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Tangents are computed with the <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span></code> member function, and the control points
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may be referenced with the <code class="computeroutput"><span class="special">.</span><span class="identifier">control_points</span></code>
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member function.
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</p>
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<p>
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The overloaded operator <span class="emphasis"><em><<</em></span> is disappointing: The
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control points are simply printed. Rendering the Bezier and its convex hull
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seems to be the best "print" statement for it, but this is essentially
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impossible in modern terminals.
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</p>
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<h4>
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<a name="math_toolkit.bezier_polynomial.h2"></a>
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<span class="phrase"><a name="math_toolkit.bezier_polynomial.caveats"></a></span><a class="link" href="bezier_polynomial.html#math_toolkit.bezier_polynomial.caveats">Caveats</a>
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</h4>
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<p>
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Do not confuse the Bezier polynomial with a Bezier spline. A Bezier spline
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has a fixed polynomial order and subdivides the curve into low-order polynomial
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||
segments. <span class="emphasis"><em>This is not a spline!</em></span> Passing <span class="emphasis"><em>n</em></span>
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control points to the <code class="computeroutput"><span class="identifier">bezier_polynomial</span></code>
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||
class creates a polynomial of degree n-1, whereas a Bezier spline has a fixed
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||
order independent of the number of control points.
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||
</p>
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<p>
|
||
Requires C++17 and support for threadlocal storage.
|
||
</p>
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<h4>
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||
<a name="math_toolkit.bezier_polynomial.h3"></a>
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<span class="phrase"><a name="math_toolkit.bezier_polynomial.performance"></a></span><a class="link" href="bezier_polynomial.html#math_toolkit.bezier_polynomial.performance">Performance</a>
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</h4>
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<p>
|
||
The following performance numbers were generated for evaluating the Bezier-polynomial.
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||
The evaluation of the interpolant is 𝑶(<span class="emphasis"><em>N</em></span>^2), as expected
|
||
from de Casteljau's algorithm.
|
||
</p>
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||
<pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="special">(</span><span class="number">16</span> <span class="identifier">X</span> <span class="number">2300</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span> <span class="identifier">s</span><span class="special">)</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">KiB</span> <span class="special">(</span><span class="identifier">x8</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">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
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<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">256</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
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<span class="identifier">L3</span> <span class="identifier">Unified</span> <span class="number">16384</span> <span class="identifier">KiB</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">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2</span> <span class="number">9.07</span> <span class="identifier">ns</span> <span class="number">9.06</span> <span class="identifier">ns</span>
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<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">3</span> <span class="number">13.2</span> <span class="identifier">ns</span> <span class="number">13.1</span> <span class="identifier">ns</span>
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<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4</span> <span class="number">17.5</span> <span class="identifier">ns</span> <span class="number">17.5</span> <span class="identifier">ns</span>
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<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">5</span> <span class="number">21.7</span> <span class="identifier">ns</span> <span class="number">21.7</span> <span class="identifier">ns</span>
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<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">6</span> <span class="number">27.4</span> <span class="identifier">ns</span> <span class="number">27.4</span> <span class="identifier">ns</span>
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<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">7</span> <span class="number">32.4</span> <span class="identifier">ns</span> <span class="number">32.3</span> <span class="identifier">ns</span>
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<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8</span> <span class="number">40.4</span> <span class="identifier">ns</span> <span class="number">40.4</span> <span class="identifier">ns</span>
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<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">9</span> <span class="number">51.9</span> <span class="identifier">ns</span> <span class="number">51.8</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">10</span> <span class="number">65.9</span> <span class="identifier">ns</span> <span class="number">65.9</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">11</span> <span class="number">79.1</span> <span class="identifier">ns</span> <span class="number">79.1</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">12</span> <span class="number">83.0</span> <span class="identifier">ns</span> <span class="number">82.9</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">13</span> <span class="number">108</span> <span class="identifier">ns</span> <span class="number">108</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">14</span> <span class="number">119</span> <span class="identifier">ns</span> <span class="number">119</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">15</span> <span class="number">140</span> <span class="identifier">ns</span> <span class="number">140</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16</span> <span class="number">137</span> <span class="identifier">ns</span> <span class="number">137</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">17</span> <span class="number">151</span> <span class="identifier">ns</span> <span class="number">151</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">18</span> <span class="number">171</span> <span class="identifier">ns</span> <span class="number">171</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">19</span> <span class="number">194</span> <span class="identifier">ns</span> <span class="number">193</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">20</span> <span class="number">213</span> <span class="identifier">ns</span> <span class="number">213</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">21</span> <span class="number">220</span> <span class="identifier">ns</span> <span class="number">220</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">22</span> <span class="number">260</span> <span class="identifier">ns</span> <span class="number">260</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">23</span> <span class="number">266</span> <span class="identifier">ns</span> <span class="number">266</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">24</span> <span class="number">293</span> <span class="identifier">ns</span> <span class="number">292</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">25</span> <span class="number">319</span> <span class="identifier">ns</span> <span class="number">319</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">26</span> <span class="number">336</span> <span class="identifier">ns</span> <span class="number">335</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">27</span> <span class="number">370</span> <span class="identifier">ns</span> <span class="number">370</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">28</span> <span class="number">429</span> <span class="identifier">ns</span> <span class="number">429</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">29</span> <span class="number">443</span> <span class="identifier">ns</span> <span class="number">443</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">30</span> <span class="number">421</span> <span class="identifier">ns</span> <span class="number">421</span> <span class="identifier">ns</span>
|
||
<span class="identifier">BezierPolynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">0.52</span> <span class="identifier">N</span><span class="special">^</span><span class="number">2</span> <span class="number">0.51</span> <span class="identifier">N</span><span class="special">^</span><span class="number">2</span>
|
||
</pre>
|
||
<p>
|
||
The Casteljau recurrence is indeed quadratic in the number of control points,
|
||
and is chosen for numerical stability. See <span class="emphasis"><em>Bezier and B-spline Techniques</em></span>,
|
||
section 2.3 for a method to Hornerize the Berstein polynomials and perhaps
|
||
produce speedups.
|
||
</p>
|
||
<h4>
|
||
<a name="math_toolkit.bezier_polynomial.h4"></a>
|
||
<span class="phrase"><a name="math_toolkit.bezier_polynomial.point_types"></a></span><a class="link" href="bezier_polynomial.html#math_toolkit.bezier_polynomial.point_types">Point
|
||
types</a>
|
||
</h4>
|
||
<p>
|
||
The <code class="computeroutput"><span class="identifier">Point</span></code> type must satisfy
|
||
certain conceptual requirements which are discussed in the documentation of
|
||
the Catmull-Rom curve. However, we reiterate them here:
|
||
</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>
|
||
</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>.
|
||
</p>
|
||
<h4>
|
||
<a name="math_toolkit.bezier_polynomial.h5"></a>
|
||
<span class="phrase"><a name="math_toolkit.bezier_polynomial.references"></a></span><a class="link" href="bezier_polynomial.html#math_toolkit.bezier_polynomial.references">References</a>
|
||
</h4>
|
||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||
<li class="listitem">
|
||
Rainer Kress, <span class="emphasis"><em>Numerical Analysis</em></span>, Springer, 1998
|
||
</li>
|
||
<li class="listitem">
|
||
David Salomon, <span class="emphasis"><em>Curves and Surfaces for Computer Graphics</em></span>,
|
||
Springer, 2005
|
||
</li>
|
||
<li class="listitem">
|
||
Prautzsch, Hartmut, Wolfgang Boehm, and Marco Paluszny. <span class="emphasis"><em>Bézier
|
||
and B-spline techniques</em></span>. Springer Science & Business Media,
|
||
2002.
|
||
</li>
|
||
</ul></div>
|
||
</div>
|
||
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, 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>
|
||
<hr>
|
||
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|
||
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