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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.bezier_polynomial"></a><a class="link" href="bezier_polynomial.html" title="Bezier Polynomials">Bezier Polynomials</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.bezier_polynomial.h0"></a>
<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>
</h4>
<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>
<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>
<span class="keyword">template</span><span class="special"><</span><span class="identifier">RandomAccessContainer</span><span class="special">></span>
<span class="keyword">class</span> <span class="identifier">bezier_polynomial</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<span class="special">};</span>
<span class="special">}</span>
</pre>
<h4>
<a name="math_toolkit.bezier_polynomial.h1"></a>
<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>
</h4>
<p>
Bézier polynomials are curves smooth curves which approximate a set of control
points. They are commonly used in computer-aided geometric design. A basic
usage is demonstrated below:
</p>
<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>
<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>
<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>
<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>
<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>
<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>
<span class="comment">// Interpolate at t = 0.1:</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">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>
</pre>
<p>
The support of the interpolant is [0,1], and an error message will be written
if attempting to evaluate the polynomial outside of these bounds. At least
two points must be passed; creating a polynomial of degree 1.
</p>
<p>
Control points may be modified via <code class="computeroutput"><span class="identifier">edit_control_point</span></code>,
for example:
</p>
<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>
<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>
</pre>
<p>
This replaces the last control point with <code class="computeroutput"><span class="identifier">endpoint</span></code>.
</p>
<p>
Tangents are computed with the <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span></code> member function, and the control points
may be referenced with the <code class="computeroutput"><span class="special">.</span><span class="identifier">control_points</span></code>
member function.
</p>
<p>
The overloaded operator <span class="emphasis"><em><<</em></span> is disappointing: The
control points are simply printed. Rendering the Bezier and its convex hull
seems to be the best "print" statement for it, but this is essentially
impossible in modern terminals.
</p>
<h4>
<a name="math_toolkit.bezier_polynomial.h2"></a>
<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>
</h4>
<p>
Do not confuse the Bezier polynomial with a Bezier spline. A Bezier spline
has a fixed polynomial order and subdivides the curve into low-order polynomial
segments. <span class="emphasis"><em>This is not a spline!</em></span> Passing <span class="emphasis"><em>n</em></span>
control points to the <code class="computeroutput"><span class="identifier">bezier_polynomial</span></code>
class creates a polynomial of degree n-1, whereas a Bezier spline has a fixed
order independent of the number of control points.
</p>
<p>
Requires C++17 and support for threadlocal storage.
</p>
<h4>
<a name="math_toolkit.bezier_polynomial.h3"></a>
<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>
</h4>
<p>
The following performance numbers were generated for evaluating the Bezier-polynomial.
The evaluation of the interpolant is 𝑶(<span class="emphasis"><em>N</em></span>^2), as expected
from de Casteljau's algorithm.
</p>
<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>
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">32</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
<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>
<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>
<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>
<span class="special">---------------------------------------------------------</span>
<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">CPU</span>
<span class="special">---------------------------------------------------------</span>
<span class="identifier">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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
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