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Bézier curve - Wikipedia, the free encyclopedia.html
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Bézier curve - Wikipedia, the free encyclopedia.html
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<h1 id="firstHeading" class="firstHeading" lang="en"><span dir="auto">Bézier curve</span></h1>
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Cubic Bézier curve</div>
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<p>A <b>Bézier curve</b> is a <a href="http://en.wikipedia.org/wiki/Parametric_curve" title="Parametric curve" class="mw-redirect">parametric curve</a> frequently used in <a href="http://en.wikipedia.org/wiki/Computer_graphics" title="Computer graphics">computer graphics</a> and related fields. Generalizations of Bézier curves to higher <a href="http://en.wikipedia.org/wiki/Dimension" title="Dimension" class="mw-redirect">dimensions</a> are called <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_surface" title="Bézier surface">Bézier surfaces</a>, of which the <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_triangle" title="Bézier triangle">Bézier triangle</a> is a special case.</p>
<p>In <a href="http://en.wikipedia.org/wiki/Vector_graphics" title="Vector graphics">vector graphics</a>, Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs,<sup id="cite_ref-1" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-1"><span>[</span>note 1<span>]</span></a></sup> are combinations of linked Bézier curves. Paths are not bound by the limits of <a href="http://en.wikipedia.org/wiki/Rasterized_image" title="Rasterized image" class="mw-redirect">rasterized images</a> and are intuitive to modify. Bézier curves are also used in <a href="http://en.wikipedia.org/wiki/Animation" title="Animation">animation</a> as a tool to control motion.<sup id="cite_ref-2" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-2"><span>[</span>note 2<span>]</span></a></sup></p>
<p>Bézier curves are also used in the time domain, particularly in animation and interface design, e.g., a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators or <a href="http://en.wikipedia.org/wiki/Graphical_user_interface" title="Graphical user interface">interface</a> designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.</p>
<p>The mathematical basis for Bézier curves — the <a href="http://en.wikipedia.org/wiki/Bernstein_polynomial" title="Bernstein polynomial">Bernstein polynomial</a> — has been known since 1912, but its applicability to graphics was understood half a century later. Bézier curves were widely publicized in 1962 by the <a href="http://en.wikipedia.org/wiki/France" title="France">French</a> engineer <a href="http://en.wikipedia.org/wiki/Pierre_B%C3%A9zier" title="Pierre Bézier">Pierre Bézier</a>, who used them to design <a href="http://en.wikipedia.org/wiki/Automobile" title="Automobile">automobile</a> bodies at <a href="http://en.wikipedia.org/wiki/Renault" title="Renault">Renault</a>. The study of these curves was however first developed in 1959 by mathematician <a href="http://en.wikipedia.org/wiki/Paul_de_Casteljau" title="Paul de Casteljau">Paul de Casteljau</a> using <a href="http://en.wikipedia.org/wiki/De_Casteljau%27s_algorithm" title="De Casteljau's algorithm">de Casteljau's algorithm</a>, a <a href="http://en.wikipedia.org/wiki/Numerical_stability" title="Numerical stability">numerically stable</a> method to evaluate Bézier curves, at <a href="http://en.wikipedia.org/wiki/Citro%C3%ABn" title="Citroën">Citroën</a>, another French automaker.</p>
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<h2>Contents</h2>
<span class="toctoggle"> [<a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#" class="internal" id="togglelink">hide</a>] </span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Applications"><span class="tocnumber">1</span> <span class="toctext">Applications</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Computer_graphics"><span class="tocnumber">1.1</span> <span class="toctext">Computer graphics</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Animation"><span class="tocnumber">1.2</span> <span class="toctext">Animation</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Fonts"><span class="tocnumber">1.3</span> <span class="toctext">Fonts</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-5"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Examination_of_cases"><span class="tocnumber">2</span> <span class="toctext">Examination of cases</span></a>
<ul>
<li class="toclevel-2 tocsection-6"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Linear_B.C3.A9zier_curves"><span class="tocnumber">2.1</span> <span class="toctext">Linear Bézier curves</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Quadratic_B.C3.A9zier_curves"><span class="tocnumber">2.2</span> <span class="toctext">Quadratic Bézier curves</span></a></li>
<li class="toclevel-2 tocsection-8"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Cubic_B.C3.A9zier_curves"><span class="tocnumber">2.3</span> <span class="toctext">Cubic Bézier curves</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-9"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Generalization"><span class="tocnumber">3</span> <span class="toctext">Generalization</span></a>
<ul>
<li class="toclevel-2 tocsection-10"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Recursive_definition"><span class="tocnumber">3.1</span> <span class="toctext">Recursive definition</span></a></li>
<li class="toclevel-2 tocsection-11"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Explicit_definition"><span class="tocnumber">3.2</span> <span class="toctext">Explicit definition</span></a></li>
<li class="toclevel-2 tocsection-12"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Terminology"><span class="tocnumber">3.3</span> <span class="toctext">Terminology</span></a></li>
<li class="toclevel-2 tocsection-13"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Properties"><span class="tocnumber">3.4</span> <span class="toctext">Properties</span></a></li>
<li class="toclevel-2 tocsection-14"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Derivative"><span class="tocnumber">3.5</span> <span class="toctext">Derivative</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-15"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Constructing_B.C3.A9zier_curves"><span class="tocnumber">4</span> <span class="toctext">Constructing Bézier curves</span></a>
<ul>
<li class="toclevel-2 tocsection-16"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Linear_curves"><span class="tocnumber">4.1</span> <span class="toctext">Linear curves</span></a></li>
<li class="toclevel-2 tocsection-17"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Quadratic_curves"><span class="tocnumber">4.2</span> <span class="toctext">Quadratic curves</span></a></li>
<li class="toclevel-2 tocsection-18"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Higher-order_curves"><span class="tocnumber">4.3</span> <span class="toctext">Higher-order curves</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-19"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Degree_elevation"><span class="tocnumber">5</span> <span class="toctext">Degree elevation</span></a>
<ul>
<li class="toclevel-2 tocsection-20"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Repeated_Degree_Elevation"><span class="tocnumber">5.1</span> <span class="toctext">Repeated Degree Elevation</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-21"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Polynomial_form"><span class="tocnumber">6</span> <span class="toctext">Polynomial form</span></a></li>
<li class="toclevel-1 tocsection-22"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Rational_B.C3.A9zier_curves"><span class="tocnumber">7</span> <span class="toctext">Rational Bézier curves</span></a></li>
<li class="toclevel-1 tocsection-23"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#See_also"><span class="tocnumber">8</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-24"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Notes"><span class="tocnumber">9</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-25"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#References"><span class="tocnumber">10</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-26"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#External_links"><span class="tocnumber">11</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<p></p>
<h2><span class="mw-headline" id="Applications">Applications</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=1" title="Edit section: Applications">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Computer_graphics">Computer graphics</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=2" title="Edit section: Computer graphics">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
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Bézier path in <a href="http://en.wikipedia.org/wiki/Adobe_Illustrator" title="Adobe Illustrator">Adobe Illustrator</a></div>
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<p>Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the <a href="http://en.wikipedia.org/wiki/Convex_hull" title="Convex hull">convex hull</a> of its <a href="http://en.wikipedia.org/wiki/Control_point_(mathematics)" title="Control point (mathematics)">control points</a>, the points can be graphically displayed and used to manipulate the curve intuitively. <a href="http://en.wikipedia.org/wiki/Affine_transformation" title="Affine transformation">Affine transformations</a> such as <a href="http://en.wikipedia.org/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> and <a href="http://en.wikipedia.org/wiki/Rotation" title="Rotation">rotation</a> can be applied on the curve by applying the respective transform on the control points of the curve.</p>
<p>Quadratic and cubic Bézier curves are most common. Higher degree curves are more <a href="http://en.wikipedia.org/wiki/Computationally_expensive" title="Computationally expensive" class="mw-redirect">computationally expensive</a> to evaluate. When more complex shapes are needed, low order Bézier curves are patched together, producing a <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_spline" title="Bézier spline">Bézier spline</a>. A Bézier spline is commonly referred to as a "path" in <a href="http://en.wikipedia.org/wiki/Vector_graphics" title="Vector graphics">vector graphics</a> standards (like <a href="http://en.wikipedia.org/wiki/SVG" title="SVG" class="mw-redirect">SVG</a>) and vector graphics programs (like <a href="http://en.wikipedia.org/wiki/Adobe_Illustrator" title="Adobe Illustrator">Adobe Illustrator</a>, <a href="http://en.wikipedia.org/wiki/CorelDraw" title="CorelDraw" class="mw-redirect">CorelDraw</a> and <a href="http://en.wikipedia.org/wiki/Inkscape" title="Inkscape">Inkscape</a>). To guarantee smoothness, the control point at which two curves meet must be on the line between the two control points on either side.</p>
<p>The simplest method for scan converting (<a href="http://en.wikipedia.org/wiki/Rasterisation" title="Rasterisation">rasterizing</a>) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments. However, this does not guarantee that the rasterized output looks sufficiently smooth, because the points may be spaced too far apart. Conversely it may generate too many points in areas where the curve is close to linear. A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line segment to within a small tolerance. If not, the curve is subdivided parametrically into two segments, 0 ≤ <i>t</i> ≤ 0.5 and 0.5 ≤ <i>t</i> ≤ 1, and the same procedure is applied recursively to each half. There are also forward differencing methods, but great care must be taken to analyse error propagation. Analytical methods where a <a href="http://en.wikipedia.org/wiki/Spline_(mathematics)" title="Spline (mathematics)">spline</a> is intersected with each scan line involve finding roots of cubic polynomials (for cubic splines) and dealing with multiple roots, so they are not often used in practice.</p>
<h3><span class="mw-headline" id="Animation">Animation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=3" title="Edit section: Animation">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>In animation applications, such as <a href="http://en.wikipedia.org/wiki/Adobe_Flash" title="Adobe Flash">Adobe Flash</a> and <a href="http://en.wikipedia.org/wiki/Synfig" title="Synfig">Synfig</a>, Bézier curves are used to outline, for example, movement. Users outline the wanted path in Bézier curves, and the application creates the needed frames for the object to move along the path. For 3D animation Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation.</p>
<h3><span class="mw-headline" id="Fonts">Fonts</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=4" title="Edit section: Fonts">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p><a href="http://en.wikipedia.org/wiki/TrueType" title="TrueType">TrueType</a> fonts use <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_spline" title="Bézier spline">Bézier splines</a> composed of <b>quadratic</b> Bézier curves. Modern imaging systems like <a href="http://en.wikipedia.org/wiki/PostScript" title="PostScript">PostScript</a>, <a href="http://en.wikipedia.org/wiki/Asymptote_(vector_graphics_language)" title="Asymptote (vector graphics language)">Asymptote</a>, <a href="http://en.wikipedia.org/wiki/Metafont" title="Metafont">Metafont</a>, and <a href="http://en.wikipedia.org/wiki/SVG" title="SVG" class="mw-redirect">SVG</a> use <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_spline" title="Bézier spline">Bézier splines</a> composed of <b>cubic</b> Bézier curves for drawing curved shapes. <a href="http://en.wikipedia.org/wiki/OpenType" title="OpenType">OpenType</a> fonts can use either kind, depending on the flavor of the font.</p>
<p>The internal rendering of all Bézier curves in font or vector graphics renderers will split them recursively up to the point where the curve is flat enough to be drawn as a series of linear or circular segments. The exact splitting algorithm is implementation dependent, only the flatness criteria must be respected to reach the necessary precision and to avoid non-monotonic local changes of curvature. The "smooth curve" feature of charts in <a href="http://en.wikipedia.org/wiki/Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a> also uses this algorithm.<sup id="cite_ref-3" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-3"><span>[</span>1<span>]</span></a></sup></p>
<p>Because arcs of circles and ellipses cannot be exactly represented by Bézier curves, they are first approximated by Bézier curves, which are in turn approximated by arcs of circles. This is inefficient as there exists also approximations of all Bézier curves using arcs of circles or ellipses, which can be rendered incrementally with arbitrary precision. Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves (or surfaces) into <a href="http://en.wikipedia.org/wiki/Non-uniform_rational_B-spline" title="Non-uniform rational B-spline">NURBS</a>, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition. This approach also allows preserving the curve definition under all linear or perspective 2D and 3D transforms and projections.</p>
<p>Font engines, like <a href="http://en.wikipedia.org/wiki/FreeType" title="FreeType">FreeType</a>, draw the font's curves (and lines) on a pixellated surface, in a process called <a href="http://en.wikipedia.org/wiki/Font_rasterization" title="Font rasterization">Font rasterization</a>.<sup id="cite_ref-freetype_4-0" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-freetype-4"><span>[</span>2<span>]</span></a></sup></p>
<h2><span class="mw-headline" id="Examination_of_cases">Examination of cases</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=5" title="Edit section: Examination of cases">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>A Bézier curve is defined by a set of <i>control points</i> <b>P</b><sub>0</sub> through <b>P</b><sub><i>n</i></sub>, where <i>n</i> is called its order (<i>n</i> = 1 for linear, 2 for quadratic, etc.). The first and last control points are always the end points of the curve; however, the intermediate control points (if any) generally do not lie on the curve.</p>
<h3><span class="mw-headline" id="Linear_B.C3.A9zier_curves">Linear Bézier curves</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=6" title="Edit section: Linear Bézier curves">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Given points <b>P</b><sub>0</sub> and <b>P</b><sub>1</sub>, a linear Bézier curve is simply a <a href="http://en.wikipedia.org/wiki/Straight_line" title="Straight line" class="mw-redirect">straight line</a> between those two points. The curve is given by</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t)=\mathbf{P}_0 + t(\mathbf{P}_1-\mathbf{P}_0)=(1-t)\mathbf{P}_0 + t\mathbf{P}_1 \mbox{ , } t \in [0,1]" src="./Bézier curve - Wikipedia, the free encyclopedia_files/ad90a6fecd5324dc32f75f0b19c2d684.png"></dd>
</dl>
<p>and is equivalent to <a href="http://en.wikipedia.org/wiki/Linear_interpolation" title="Linear interpolation">linear interpolation</a>.</p>
<h3><span class="mw-headline" id="Quadratic_B.C3.A9zier_curves">Quadratic Bézier curves</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=7" title="Edit section: Quadratic Bézier curves">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>A quadratic Bézier curve is the path traced by the function <b>B</b>(<i>t</i>), given points <b>P</b><sub>0</sub>, <b>P</b><sub>1</sub>, and <b>P</b><sub>2</sub>,</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t) = (1 - t)[(1 - t) \mathbf P_0 + t \mathbf P_1] + t [(1 - t) \mathbf P_1 + t \mathbf P_2] \mbox{ , } t \in [0,1]" src="./Bézier curve - Wikipedia, the free encyclopedia_files/09d743f944f43b8c8a3c8468cabdb072.png">,</dd>
</dl>
<p>which can be interpreted as the linear interpolant of corresponding points on the linear Bézier curves from <b>P</b><sub>0</sub> to <b>P</b><sub>1</sub> and from <b>P</b><sub>1</sub> to <b>P</b><sub>2</sub> respectively. Rearranging the preceding equation yields:</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t) = (1 - t)^{2}\mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^{2}\mathbf{P}_2 \mbox{ , } t \in [0,1]." src="./Bézier curve - Wikipedia, the free encyclopedia_files/2d5e5d58562d8ec2c35f16df98d2b974.png"></dd>
</dl>
<p>The derivative of the Bézier curve with respect to <i>t</i> is</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}'(t) = 2 (1 - t) (\mathbf{P}_1 - \mathbf{P}_0) + 2 t (\mathbf{P}_2 - \mathbf{P}_1) \,." src="./Bézier curve - Wikipedia, the free encyclopedia_files/e0288487bb9614866c5e8f722949a3ba.png"></dd>
</dl>
<p>from which it can be concluded that the tangents to the curve at <b>P</b><sub>0</sub> and <b>P</b><sub>2</sub> intersect at <b>P</b><sub>1</sub>. As <i>t</i> increases from 0 to 1, the curve departs from <b>P</b><sub>0</sub> in the direction of <b>P</b><sub>1</sub>, then bends to arrive at <b>P</b><sub>2</sub> from the direction of <b>P</b><sub>1</sub>.</p>
<p>The second derivative of the Bézier curve with respect to <i>t</i> is</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}''(t) = 2(\mathbf{P}_2 - 2 \mathbf{P}_1 + \mathbf{P}_0) \,." src="./Bézier curve - Wikipedia, the free encyclopedia_files/010e178bd06c630d02d1c22df4cb92fd.png"></dd>
</dl>
<p>A quadratic Bézier curve is also a <a href="http://en.wikipedia.org/wiki/Parabola" title="Parabola">parabolic</a> segment. As a parabola is a <a href="http://en.wikipedia.org/wiki/Conic_section" title="Conic section">conic section</a>, some sources refer to quadratic Béziers as "conic arcs".<sup id="cite_ref-freetype_4-1" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-freetype-4"><span>[</span>2<span>]</span></a></sup></p>
<h3><span class="mw-headline" id="Cubic_B.C3.A9zier_curves">Cubic Bézier curves</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=8" title="Edit section: Cubic Bézier curves">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Four points <b>P</b><sub>0</sub>, <b>P</b><sub>1</sub>, <b>P</b><sub>2</sub> and <b>P</b><sub>3</sub> in the plane or in higher-dimensional space define a cubic Bézier curve. The curve starts at <b>P</b><sub>0</sub> going toward <b>P</b><sub>1</sub> and arrives at <b>P</b><sub>3</sub> coming from the direction of <b>P</b><sub>2</sub>. Usually, it will not pass through <b>P</b><sub>1</sub> or <b>P</b><sub>2</sub>; these points are only there to provide directional information. The distance between <b>P</b><sub>0</sub> and <b>P</b><sub>1</sub> determines "how long" the curve moves into direction <b>P</b><sub>2</sub> before turning towards <b>P</b><sub>3</sub>.</p>
<p>Writing <b>B</b><sub><b>P</b><sub>i</sub>,<b>P</b><sub>j</sub>,<b>P</b><sub>k</sub></sub>(<i>t</i>) for the quadratic Bézier curve defined by points <b>P</b><sub>i</sub>, <b>P</b><sub>j</sub>, and <b>P</b><sub>k</sub>, the cubic Bézier curve can be defined as a linear combination of two quadratic Bézier curves:</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t)=(1-t)\mathbf{B}_{\mathbf P_0,\mathbf P_1,\mathbf P_2}(t) + t \mathbf{B}_{\mathbf P_1,\mathbf P_2,\mathbf P_3}(t) \mbox{ , } t \in [0,1]." src="./Bézier curve - Wikipedia, the free encyclopedia_files/fdbeda5f0c7bbea16e80663c606ba863.png"></dd>
</dl>
<p>The explicit form of the curve is:</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t)=(1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3 \mbox{ , } t \in [0,1]." src="./Bézier curve - Wikipedia, the free encyclopedia_files/5e32b674a98a9f70e492851f9ad92b61.png"></dd>
</dl>
<p>For some choices of <b>P</b><sub>1</sub> and <b>P</b><sub>2</sub> the curve may intersect itself, or contain a cusp.</p>
<p>Any series of any 4 distinct points can be converted to a cubic Bézier curve that goes through all 4 points in order. Given the starting and ending point of some cubic Bézier curve, and the points along the curve corresponding to <i>t</i> = 1/3 and <i>t</i> = 2/3, the control points for the original Bézier curve can be recovered.<sup id="cite_ref-5" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-5"><span>[</span>3<span>]</span></a></sup></p>
<p>The derivative of the cubic Bézier curve with respect to <i>t</i> is</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}'(t) = 3(1-t)^2(\mathbf{P}_1 - \mathbf{P}_0) + 6(1-t)t(\mathbf{P}_2 - \mathbf{P}_1) + 3t^2(\mathbf{P}_3 - \mathbf{P}_2) \,." src="./Bézier curve - Wikipedia, the free encyclopedia_files/0e6e805a66780eabc7988903aa14dd1a.png"></dd>
</dl>
<p><br>
The second derivative of the Bézier curve with respect to <i>t</i> is</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}''(t) = 6(1-t)(\mathbf{P}_2 - 2 \mathbf{P}_1 + \mathbf{P}_0) + 6t(\mathbf{P}_3 - 2 \mathbf{P}_2 + \mathbf{P}_1) \,." src="./Bézier curve - Wikipedia, the free encyclopedia_files/f2b0fd1acf189ce110a1f41aa2d697fa.png"></dd>
</dl>
<h2><span class="mw-headline" id="Generalization">Generalization</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=9" title="Edit section: Generalization">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Bézier curves can be defined for any degree <i>n.</i></p>
<h3><span class="mw-headline" id="Recursive_definition">Recursive definition</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=10" title="Edit section: Recursive definition">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>A recursive definition for the Bézier curve of degree <i>n</i> expresses it as a point-to-point <a href="http://en.wikipedia.org/wiki/Linear_combination" title="Linear combination">linear combination</a> (<a href="http://en.wikipedia.org/wiki/Linear_interpolation" title="Linear interpolation">linear interpolation</a>) of a pair of corresponding points in two Bézier curves of degree <i>n</i> − 1.</p>
<p>Let <img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_n}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/12482104eb5115c14219436749bcbbc0.png"> denote the Bézier curve determined by any selection of points <b>P</b><sub>0</sub>, <b>P</b><sub>1</sub>, ..., <b>P</b><sub><i>n</i></sub>. Then to start,</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}_{\mathbf{P}_0}(t) = \mathbf{P}_0 \text{, and}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/9c7efd6edd45de7cf33716b4e67ebe3b.png"></dd>
</dl>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t) = \mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_n}(t) = (1-t)\mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_{n-1}}(t) + t\mathbf{B}_{\mathbf{P}_1\mathbf{P}_2\ldots\mathbf{P}_n}(t)" src="./Bézier curve - Wikipedia, the free encyclopedia_files/99a7fa8da2d1abbcb3fb8a5723eb75bb.png"></dd>
</dl>
<p>This recursion is elucidated in the animations below.</p>
<h3><span class="mw-headline" id="Explicit_definition">Explicit definition</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=11" title="Edit section: Explicit definition">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The formula can be expressed explicitly as follows:</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\begin{align}
\mathbf{B}(t) = {} &\sum_{i=0}^n {n\choose i}(1 - t)^{n - i}t^i\mathbf{P}_i \\
= {} &(1 - t)^n\mathbf{P}_0 + {n\choose 1}(1 - t)^{n - 1}t\mathbf{P}_1 + \cdots \\
{} &\cdots + {n\choose n - 1}(1 - t)t^{n - 1}\mathbf{P}_{n - 1} + t^n\mathbf{P}_n,\quad t \in [0,1]
\end{align}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/fdd94eee9ac9bd8ef46a0c8fbb003e69.png"></dd>
</dl>
<p>where <img class="mwe-math-fallback-png-inline tex" alt="\scriptstyle {n \choose i}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/3197349f5acc1a0108fba95667bcecfd.png"> are the <a href="http://en.wikipedia.org/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>.</p>
<p>For example, for <i>n</i> = 5:</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\begin{align}
\mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\mathbf{P}_2\mathbf{P}_3\mathbf{P}_4\mathbf{P}_5}(t) = \mathbf{B}(t)
= {} & (1 - t)^5\mathbf{P}_0 + 5t(1 - t)^4\mathbf{P}_1 + 10t^2(1 - t)^3 \mathbf{P}_2 \\
{} & + 10t^3 (1-t)^2 \mathbf{P}_3 + 5t^4(1-t) \mathbf{P}_4 + t^5 \mathbf{P}_5,\quad t \in [0,1]
\end{align}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/aa69c29675437a20bcb28fb68e6b126e.png"></dd>
</dl>
<h3><span class="mw-headline" id="Terminology">Terminology</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=12" title="Edit section: Terminology">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Some terminology is associated with these parametric curves. We have</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t) = \sum_{i=0}^n b_{i, n}(t)\mathbf{P}_i,\quad t \in [0, 1]" src="./Bézier curve - Wikipedia, the free encyclopedia_files/4f781b246afadfaf93fdce53fadc753b.png"></dd>
</dl>
<p>where the polynomials</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="b_{i,n}(t) = {n\choose i} t^i (1 - t)^{n - i},\quad i = 0, \ldots, n" src="./Bézier curve - Wikipedia, the free encyclopedia_files/1372049a52b9c2e6b7fa2d9ba35a3679.png"></dd>
</dl>
<p>are known as <a href="http://en.wikipedia.org/wiki/Bernstein_polynomial" title="Bernstein polynomial">Bernstein basis polynomials</a> of degree <i>n</i>.</p>
<p>Note that <i>t</i><sup>0</sup> = 1, (1 − <i>t</i>)<sup>0</sup> = 1, and that the <a href="http://en.wikipedia.org/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>, <img class="mwe-math-fallback-png-inline tex" alt="\scriptstyle {n \choose i}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/3197349f5acc1a0108fba95667bcecfd.png">, also expressed as <img class="mwe-math-fallback-png-inline tex" alt="^n{\mathbf{C}}_i " src="./Bézier curve - Wikipedia, the free encyclopedia_files/78963a9902a193f7f9553cee91b26e33.png"> or <img class="mwe-math-fallback-png-inline tex" alt="{\mathbf{C}_i}^n " src="./Bézier curve - Wikipedia, the free encyclopedia_files/8893a70f26799a17e3cdb51f8d5fcf6a.png"> is:</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="{n \choose i} = \frac{n!}{i!(n - i)!}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/4e839409fce004a96b91d2be0d8d931d.png"></dd>
</dl>
<p>The points <b>P</b><sub><i>i</i></sub> are called <i>control points</i> for the Bézier curve. The <a href="http://en.wikipedia.org/wiki/Polygon" title="Polygon">polygon</a> formed by connecting the Bézier points with <a href="http://en.wikipedia.org/wiki/Line_(mathematics)" title="Line (mathematics)" class="mw-redirect">lines</a>, starting with <b>P</b><sub>0</sub> and finishing with <b>P</b><sub><i>n</i></sub>, is called the <i>Bézier polygon</i> (or <i>control polygon</i>). The <a href="http://en.wikipedia.org/wiki/Convex_hull" title="Convex hull">convex hull</a> of the Bézier polygon contains the Bézier curve.</p>
<h3><span class="mw-headline" id="Properties">Properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=13" title="Edit section: Properties">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<ul>
<li>The curve begins at <b>P</b><sub>0</sub> and ends at <b>P</b><sub><i>n</i></sub>; this is the so-called <i>endpoint interpolation</i> property.</li>
<li>The curve is a straight line if and only if all the control points are <a href="http://en.wikipedia.org/wiki/Incidence_(geometry)#Collinearity" title="Incidence (geometry)">collinear</a>.</li>
<li>The start (end) of the curve is <a href="http://en.wikipedia.org/wiki/Tangent" title="Tangent">tangent</a> to the first (last) section of the Bézier polygon.</li>
<li>A curve can be split at any point into two subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.</li>
<li>Some curves that seem simple, such as the <a href="http://en.wikipedia.org/wiki/Circle" title="Circle">circle</a>, cannot be described exactly by a Bézier or <a href="http://en.wikipedia.org/wiki/Piecewise" title="Piecewise">piecewise</a> Bézier curve; though a four-piece cubic Bézier curve can approximate a circle (see <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_spline" title="Bézier spline">Bézier spline</a>), with a maximum radial error of less than one part in a thousand, when each inner control point (or offline point) is the distance <img class="mwe-math-fallback-png-inline tex" alt="\textstyle\frac{4\left(\sqrt {2}-1\right)}{3}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/bcf5bd258d8efefb0dad147d58748ecc.png"> horizontally or vertically from an outer control point on a unit circle. More generally, an <i>n</i>-piece cubic Bézier curve can approximate a circle, when each inner control point is the distance <img class="mwe-math-fallback-png-inline tex" alt="\textstyle\frac{4}{3}\tan(t/4)" src="./Bézier curve - Wikipedia, the free encyclopedia_files/3aa02425732dbc4757f74a5d2156ecb0.png"> from an outer control point on a unit circle, where <i>t</i> is 360/<i>n</i> degrees, and <i>n</i> > 2.</li>
<li>The curve at a fixed offset from a given Bézier curve, often called an <i>offset curve</i> (lying "parallel" to the original curve, like the offset between rails in a <a href="http://en.wikipedia.org/wiki/Railroad_track" title="Railroad track" class="mw-redirect">railroad track</a>), cannot be exactly formed by a Bézier curve (except in some trivial cases). However, there are <a href="http://en.wikipedia.org/wiki/Heuristic" title="Heuristic">heuristic</a> methods that usually give an adequate approximation for practical purposes. (For example: <a rel="nofollow" class="external autonumber" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.1724">[1]</a>)</li>
<li>Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree <i>n</i> Bézier curve is also a degree <i>m</i> curve for any <i>m</i> > <i>n</i>. In detail, a degree <i>n</i> curve with control points <b>P</b><sub>0</sub>, …, <b>P</b><sub><i>n</i></sub> is equivalent (including the parametrization) to the degree <i>n</i> + 1 curve with control points <b>P'</b><sub>0</sub>, …, <b>P'</b><sub><i>n</i> + 1</sub>, where <img class="mwe-math-fallback-png-inline tex" alt="\mathbf P'_k=\tfrac{k}{n+1}\mathbf P_{k-1}+\left(1-\tfrac{k}{n+1}\right)\mathbf P_k" src="./Bézier curve - Wikipedia, the free encyclopedia_files/7b82aad853ef84aae944bf14d4bed641.png">.</li>
<li>Bézier curves follow the <a href="http://en.wikipedia.org/w/index.php?title=Variation_Diminishing_Property&action=edit&redlink=1" class="new" title="Variation Diminishing Property (page does not exist)">Variation Diminishing Property</a>.</li>
</ul>
<h3><span class="mw-headline" id="Derivative">Derivative</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=14" title="Edit section: Derivative">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The generalised derivative for a curve of order <i>n</i> is</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}'(t) = n \sum_{i=0}^{n-1} b_{i,n-1}(t) (\mathbf{P}_{i+1} - \mathbf{P}_i)" src="./Bézier curve - Wikipedia, the free encyclopedia_files/2a155c7c1fade846c3315ec855bf26ce.png"></dd>
</dl>
<h2><span class="mw-headline" id="Constructing_B.C3.A9zier_curves">Constructing Bézier curves</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=15" title="Edit section: Constructing Bézier curves">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Linear_curves">Linear curves</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=16" title="Edit section: Linear curves">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<table style="text-align:center; float:right; font-size:95%;">
<tbody><tr>
<td style="border-bottom: 1px solid #222222;"><a href="http://en.wikipedia.org/wiki/File:B%C3%A9zier_1_big.gif" class="image" title="Animation of a linear Bézier curve, t in [0,1]"><img alt="Animation of a linear Bézier curve, t in [0,1]" src="./Bézier curve - Wikipedia, the free encyclopedia_files/240px-Bézier_1_big.gif" width="240" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/0/00/B%C3%A9zier_1_big.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/00/B%C3%A9zier_1_big.gif 2x" data-file-width="360" data-file-height="150"></a></td>
</tr>
<tr>
<td>Animation of a linear Bézier curve, <i>t</i> in [0,1]</td>
</tr>
</tbody></table>
<p>The <i>t</i> in the function for a linear Bézier curve can be thought of as describing how far <b>B</b>(<i>t</i>) is from <b>P</b><sub>0</sub> to <b>P</b><sub>1</sub>. For example when <i>t=0.25</i>, <b>B</b>(<i>t</i>) is one quarter of the way from point <b>P</b><sub>0</sub> to <b>P</b><sub>1</sub>. As <i>t</i> varies from 0 to 1, <b>B</b>(<i>t</i>) describes a straight line from <b>P</b><sub>0</sub> to <b>P</b><sub>1</sub>.</p>
<h3><span class="mw-headline" id="Quadratic_curves">Quadratic curves</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=17" title="Edit section: Quadratic curves">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>For quadratic Bézier curves one can construct intermediate points <b>Q</b><sub>0</sub> and <b>Q</b><sub>1</sub> such that as <i>t</i> varies from 0 to 1:</p>
<ul>
<li>Point <b>Q</b><sub>0</sub> varies from <b>P</b><sub>0</sub> to <b>P</b><sub>1</sub> and describes a linear Bézier curve.</li>
<li>Point <b>Q</b><sub>1</sub> varies from <b>P</b><sub>1</sub> to <b>P</b><sub>2</sub> and describes a linear Bézier curve.</li>
<li>Point <b>B</b>(<i>t</i>) varies from <b>Q</b><sub>0</sub> to <b>Q</b><sub>1</sub> and describes a quadratic Bézier curve.</li>
</ul>
<center>
<table style="text-align:center; float:none; clear:both; font-size:95%;">
<tbody><tr>
<td style="border-bottom: 1px solid #22ff22;"><a href="http://en.wikipedia.org/wiki/File:B%C3%A9zier_2_big.svg" class="image" title="Construction of a quadratic Bézier curve"><img alt="Construction of a quadratic Bézier curve" src="./Bézier curve - Wikipedia, the free encyclopedia_files/240px-Bézier_2_big.svg.png" width="240" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/B%C3%A9zier_2_big.svg/360px-B%C3%A9zier_2_big.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/B%C3%A9zier_2_big.svg/480px-B%C3%A9zier_2_big.svg.png 2x" data-file-width="360" data-file-height="150"></a></td>
<td></td>
<td style="border-bottom: 1px solid #22ff22;"><a href="http://en.wikipedia.org/wiki/File:B%C3%A9zier_2_big.gif" class="image" title="Animation of a quadratic Bézier curve, t in [0,1]"><img alt="Animation of a quadratic Bézier curve, t in [0,1]" src="./Bézier curve - Wikipedia, the free encyclopedia_files/240px-Bézier_2_big.gif" width="240" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/3/3d/B%C3%A9zier_2_big.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3d/B%C3%A9zier_2_big.gif 2x" data-file-width="360" data-file-height="150"></a></td>
</tr>
<tr>
<td>Construction of a quadratic Bézier curve</td>
<td></td>
<td>Animation of a quadratic Bézier curve, <i>p</i> in [0,1]</td>
</tr>
</tbody></table>
</center>
<h3><span class="mw-headline" id="Higher-order_curves">Higher-order curves</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=18" title="Edit section: Higher-order curves">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>For higher-order curves one needs correspondingly more intermediate points. For cubic curves one can construct intermediate points <b>Q</b><sub>0</sub>, <b>Q</b><sub>1</sub>, and <b>Q</b><sub>2</sub> that describe linear Bézier curves, and points <b>R</b><sub>0</sub> & <b>R</b><sub>1</sub> that describe quadratic Bézier curves:</p>
<center>
<table style="text-align:center; float:none; clear:both; font-size:95%;">
<tbody><tr>
<td style="border-bottom: 1px solid #2222ff"><a href="http://en.wikipedia.org/wiki/File:B%C3%A9zier_3_big.svg" class="image" title="Construction of a cubic Bézier curve"><img alt="Construction of a cubic Bézier curve" src="./Bézier curve - Wikipedia, the free encyclopedia_files/240px-Bézier_3_big.svg.png" width="240" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/B%C3%A9zier_3_big.svg/360px-B%C3%A9zier_3_big.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/B%C3%A9zier_3_big.svg/480px-B%C3%A9zier_3_big.svg.png 2x" data-file-width="360" data-file-height="150"></a></td>
<td></td>
<td style="border-bottom: 1px solid #2222ff"><a href="http://en.wikipedia.org/wiki/File:B%C3%A9zier_3_big.gif" class="image" title="Animation of a cubic Bézier curve, t in [0,1]"><img alt="Animation of a cubic Bézier curve, t in [0,1]" src="./Bézier curve - Wikipedia, the free encyclopedia_files/240px-Bézier_3_big.gif" width="240" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/d/db/B%C3%A9zier_3_big.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/d/db/B%C3%A9zier_3_big.gif 2x" data-file-width="360" data-file-height="150"></a></td>
</tr>
<tr>
<td>Construction of a cubic Bézier curve</td>
<td></td>
<td>Animation of a cubic Bézier curve, <i>t</i> in [0,1]</td>
</tr>
</tbody></table>
</center>
<p>For fourth-order curves one can construct intermediate points <b>Q</b><sub>0</sub>, <b>Q</b><sub>1</sub>, <b>Q</b><sub>2</sub> & <b>Q</b><sub>3</sub> that describe linear Bézier curves, points <b>R</b><sub>0</sub>, <b>R</b><sub>1</sub> & <b>R</b><sub>2</sub> that describe quadratic Bézier curves, and points <b>S</b><sub>0</sub> & <b>S</b><sub>1</sub> that describe cubic Bézier curves:</p>
<center>
<table style="text-align:center; float:none; clear:both; font-size:95%;">
<tbody><tr>
<td style="border-bottom: 1px solid #ff22ff"><a href="http://en.wikipedia.org/wiki/File:B%C3%A9zier_4_big.svg" class="image" title="Construction of a quartic Bézier curve"><img alt="Construction of a quartic Bézier curve" src="./Bézier curve - Wikipedia, the free encyclopedia_files/240px-Bézier_4_big.svg.png" width="240" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/B%C3%A9zier_4_big.svg/360px-B%C3%A9zier_4_big.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/B%C3%A9zier_4_big.svg/480px-B%C3%A9zier_4_big.svg.png 2x" data-file-width="360" data-file-height="150"></a></td>
<td></td>
<td style="border-bottom: 1px solid #ff22ff"><a href="http://en.wikipedia.org/wiki/File:B%C3%A9zier_4_big.gif" class="image" title="Animation of a quartic Bézier curve, t in [0,1]"><img alt="Animation of a quartic Bézier curve, t in [0,1]" src="./Bézier curve - Wikipedia, the free encyclopedia_files/240px-Bézier_4_big.gif" width="240" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/a/a4/B%C3%A9zier_4_big.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a4/B%C3%A9zier_4_big.gif 2x" data-file-width="360" data-file-height="150"></a></td>
</tr>
<tr>
<td>Construction of a quartic Bézier curve</td>
<td></td>
<td>Animation of a quartic Bézier curve, <i>t</i> in [0,1]</td>
</tr>
</tbody></table>
</center>
<p>For fifth-order curves, one can construct similar intermediate points.</p>
<center>
<table style="text-align:center; float:none; clear:both; font-size:95%">
<tbody><tr>
<td style="border-bottom: 1px solid silver"><a href="http://en.wikipedia.org/wiki/File:BezierCurve.gif" class="image" title="Animation of the construction of a fifth-order Bézier curve"><img alt="Animation of the construction of a fifth-order Bézier curve" src="./Bézier curve - Wikipedia, the free encyclopedia_files/240px-BezierCurve.gif" width="240" height="192" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/BezierCurve.gif/360px-BezierCurve.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0b/BezierCurve.gif 2x" data-file-width="473" data-file-height="378"></a></td>
</tr>
<tr>
<td>Animation of a fifth order Bézier curve, <i>t</i> in [0,1]</td>
</tr>
</tbody></table>
</center>
<p>These representations rest on the process used in <a href="http://en.wikipedia.org/wiki/De_Casteljau%27s_algorithm" title="De Casteljau's algorithm">De Casteljau's algorithm</a> to calculate Bézier curves.<sup id="cite_ref-6" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-6"><span>[</span>4<span>]</span></a></sup></p>
<h2><span class="mw-headline" id="Degree_elevation">Degree elevation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=19" title="Edit section: Degree elevation">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>A Bézier curve of degree <i>n</i> can be converted into a Bézier curve of degree <i>n</i> + 1 <i>with the same shape</i>. This is useful if software supports Bézier curves only of specific degree. For example, you can draw a quadratic Bézier curve with <a href="http://en.wikipedia.org/wiki/Cairo_(graphics)" title="Cairo (graphics)">Cairo</a>, which supports only cubic Bézier curves.</p>
<p>To do degree elevation, we use the equality <img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t) = (1-t)\mathbf{B}(t) + t\mathbf{B}(t)" src="./Bézier curve - Wikipedia, the free encyclopedia_files/6560423d6fc918446719e0c5bc8d9258.png">. Each component <img class="mwe-math-fallback-png-inline tex" alt="\mathbf{b}_{i,n}(t)\mathbf{P}_i" src="./Bézier curve - Wikipedia, the free encyclopedia_files/9370e612c32406e6572ed5aead0a2a5f.png"> is multiplied by (1 − <i>t</i>) or <i>t</i>, thus increasing a degree by one. Here is the example of increasing degree from 2 to 3.</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="
\begin{align}
&(1 - t)^2 \mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^2 \mathbf{P}_2 \\
= {} &(1 - t)^3 \mathbf{P}_0 + (1 - t)^{2}t\mathbf{P}_0 + 2(1 - t)^2 t\mathbf{P}_1 \\
&+ 2(1 - t)t^2 \mathbf{P}_1 + (1 - t)t^2 \mathbf{P}_2 + t^3 \mathbf{P}_2 \\
= {} &(1 - t)^3 \mathbf{P}_0
+ (1 - t)^2 t \left( \mathbf{P}_0 + 2\mathbf{P}_1\right)
+ (1 - t) t^2 \left(2\mathbf{P}_1 + \mathbf{P}_2\right)
+ t^{3}\mathbf{P}_2 \\
= {} &(1 - t)^3 \mathbf{P}_0
+ 3(1 - t)^2 t \left( \frac{\mathbf{P}_0 + 2\mathbf{P}_1}{3} \right)
+ 3(1 - t) t^2 \left( \frac{2\mathbf{P}_1 + \mathbf{P}_2}{3} \right)
+ t^{3}\mathbf{P}_2
\end{align}
" src="./Bézier curve - Wikipedia, the free encyclopedia_files/42047f3adec8f4a16788efd4729a2b32.png"></dd>
</dl>
<p>For arbitrary <i>n</i> we use equalities</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\begin{align}
{n + 1 \choose i}(1 - t)\mathbf{b}_{i, n} &= {n \choose i} \mathbf{b}_{i, n + 1} \Rightarrow (1 - t)\mathbf{b}_{i, n} = \frac{n + 1 - i}{n + 1} \mathbf{b}_{i, n + 1} \\
{n + 1 \choose i + 1} t\mathbf{b}_{i, n} &= {n \choose i} \mathbf{b}_{i + 1, n + 1} \Rightarrow t\mathbf{b}_{i, n} = \frac{i + 1}{n + 1} \mathbf{b}_{i + 1, n + 1} \\
\mathbf{B}(t) &= (1 - t)\sum_{i=0}^n \mathbf{b}_{i, n}(t)\mathbf{P}_i
+ t\sum_{i=0}^n \mathbf{b}_{i, n}(t)\mathbf{P}_i \\
&= \sum_{i=0}^n \frac{n + 1 - i}{n + 1}\mathbf{b}_{i, n + 1}(t)\mathbf{P}_i
+ \sum_{i=0}^n \frac{i + 1}{n + 1}\mathbf{b}_{i + 1, n + 1}(t)\mathbf{P}_i \\
&= \sum_{i=0}^{n + 1} \left(\frac{i}{n + 1}\mathbf{P}_{i - 1}
+ \frac{n + 1 - i}{n + 1}\mathbf{P}_i\right) \mathbf{b}_{i, n + 1}(t) \\
&= \sum_{i=0}^{n+1} \mathbf{b}_{i, n + 1}(t)\mathbf{P'}_i
\end{align}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/e7378fcfcf9f8034d61a4ab6ffb6c98e.png"></dd>
</dl>
<p>introducing arbitrary <img class="mwe-math-fallback-png-inline tex" alt="\mathbf{P}_{-1}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/ee6ef2065ef681d00035f4ad7d370785.png"> and <img class="mwe-math-fallback-png-inline tex" alt="\mathbf{P}_{n + 1}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/af5a0e23fd65c7b6a652d16b2a8674a3.png">.</p>
<p>Therefore new control points are <sup id="cite_ref-7" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-7"><span>[</span>5<span>]</span></a></sup></p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{P'}_i = \frac{i}{n + 1}\mathbf{P}_{i - 1} + \frac{n + 1 - i}{n + 1}\mathbf{P}_i,\quad i=0, \ldots, n + 1" src="./Bézier curve - Wikipedia, the free encyclopedia_files/cd77b1e57fa46b4012be933bcebfad5d.png"></dd>
</dl>
<h3><span class="mw-headline" id="Repeated_Degree_Elevation">Repeated Degree Elevation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=20" title="Edit section: Repeated Degree Elevation">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The concept of Degree Elevation can be repeated on a control polygon <b>R</b> to get a sequence of control polygons <b>R</b>,<b>R</b><sub>1</sub>,<b>R</b><sub>2</sub>, and so on. After <i>r</i> degree elevations, the polygon <b>R</b><sub>r</sub> has the vertices <b>P</b><sub>0,r</sub>,<b>P</b><sub>1,r</sub>,<b>P</b><sub>2,r</sub>,...,<b>P</b><sub>n+r,r</sub> given by <sup id="cite_ref-8" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-8"><span>[</span>6<span>]</span></a></sup></p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{P}_{i,r} = \sum_{j=0}^n \mathbf{P}_j \tbinom nj \frac{\tbinom{r}{i-j}}{\tbinom{n+r}{i}}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/b0ae179b3dc9b311910e8ece82b1b804.png"></dd>
</dl>
<p>It can also be shown that for the underlying Bézier curve <i>B</i>,</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="\mathbf{\lim_{r \to \infty}R_r} = \mathbf{B}" src="./Bézier curve - Wikipedia, the free encyclopedia_files/0682120e2628f8bf5d74a3d58e092457.png"></dd>
</dl>
<h2><span class="mw-headline" id="Polynomial_form">Polynomial form</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=21" title="Edit section: Polynomial form">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Sometimes it is desirable to express the Bézier curve as a <a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial">polynomial</a> instead of a sum of less straightforward <a href="http://en.wikipedia.org/wiki/Bernstein_polynomial" title="Bernstein polynomial">Bernstein polynomials</a>. Application of the <a href="http://en.wikipedia.org/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a> to the definition of the curve followed by some rearrangement will yield:</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="
\mathbf{B}(t) = \sum_{j = 0}^n {t^j \mathbf{C}_j}
" src="./Bézier curve - Wikipedia, the free encyclopedia_files/e970f51b996903c7d470c0bcecd6f22e.png"></dd>
</dl>
<p>where</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="
\mathbf{C}_j = \frac{n!}{(n - j)!} \sum_{i = 0}^j \frac{(-1)^{i + j} \mathbf{P}_i}{i! (j - i)!} =
\prod_{m = 0}^{j - 1} (n - m) \sum_{i = 0}^j \frac{(-1)^{i + j} \mathbf{P}_i}{i! (j - i)!}
." src="./Bézier curve - Wikipedia, the free encyclopedia_files/83893d1f4494d4e2cc8e84284400b319.png"></dd>
</dl>
<p>This could be practical if <img class="mwe-math-fallback-png-inline tex" alt="\mathbf{C}_j" src="./Bézier curve - Wikipedia, the free encyclopedia_files/759702f7ef195cde6bad3df9c9b5975e.png"> can be computed prior to many evaluations of <img class="mwe-math-fallback-png-inline tex" alt="\mathbf{B}(t)" src="./Bézier curve - Wikipedia, the free encyclopedia_files/7bd948ee96cd5c49bf6a9c6e879e80c7.png">; however one should use caution as high order curves may lack <a href="http://en.wikipedia.org/wiki/Numeric_stability" title="Numeric stability" class="mw-redirect">numeric stability</a> (<a href="http://en.wikipedia.org/wiki/De_Casteljau%27s_algorithm" title="De Casteljau's algorithm">de Casteljau's algorithm</a> should be used if this occurs). Note that the <a href="http://en.wikipedia.org/wiki/Empty_product" title="Empty product">empty product</a> is 1.</p>
<h2><span class="mw-headline" id="Rational_B.C3.A9zier_curves">Rational Bézier curves</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=22" title="Edit section: Rational Bézier curves">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:Rational_Bezier_curve-conic_sections.svg" class="image"><img alt="" src="./Bézier curve - Wikipedia, the free encyclopedia_files/220px-Rational_Bezier_curve-conic_sections.svg.png" width="220" height="168" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Rational_Bezier_curve-conic_sections.svg/330px-Rational_Bezier_curve-conic_sections.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Rational_Bezier_curve-conic_sections.svg/440px-Rational_Bezier_curve-conic_sections.svg.png 2x" data-file-width="380" data-file-height="290"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Rational_Bezier_curve-conic_sections.svg" class="internal" title="Enlarge"><img src="./Bézier curve - Wikipedia, the free encyclopedia_files/magnify-clip.png" width="15" height="11" alt=""></a></div>
Sections of conic sections represented exactly by rational Bézier curves</div>
</div>
</div>
<p>The rational Bézier curve adds adjustable weights to provide closer approximations to arbitrary shapes. The numerator is a weighted Bernstein-form Bézier curve and the denominator is a weighted sum of <a href="http://en.wikipedia.org/wiki/Bernstein_polynomial" title="Bernstein polynomial">Bernstein polynomials</a>. Rational Bézier curves can, among other uses, be used to represent segments of <a href="http://en.wikipedia.org/wiki/Conic_section" title="Conic section">conic sections</a> exactly.<sup id="cite_ref-9" class="reference"><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_note-9"><span>[</span>7<span>]</span></a></sup></p>
<p>Given <i>n</i> + 1 control points <b>P</b><sub><i>i</i></sub>, the rational Bézier curve can be described by:</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="
\mathbf{B}(t) =
\frac{
\sum_{i=0}^n b_{i,n}(t) \mathbf{P}_{i}w_i
}
{
\sum_{i=0}^n b_{i,n}(t) w_i
}
" src="./Bézier curve - Wikipedia, the free encyclopedia_files/0d8fa1beb8c7c4e30be9fd199e993ffc.png"></dd>
</dl>
<p>or simply</p>
<dl>
<dd><img class="mwe-math-fallback-png-inline tex" alt="
\mathbf{B}(t) =
\frac{
\sum_{i=0}^n {n \choose i} t^i (1-t)^{n-i}\mathbf{P}_{i}w_i
}
{
\sum_{i=0}^n {n \choose i} t^i (1-t)^{n-i}w_i
}.
" src="./Bézier curve - Wikipedia, the free encyclopedia_files/3025ba39155c31511fcc675edf2f4621.png"></dd>
</dl>
<h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=23" title="Edit section: See also">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:102px;"><a href="http://en.wikipedia.org/wiki/File:Quadratic_Beziers_in_string_art.svg" class="image"><img alt="" src="./Bézier curve - Wikipedia, the free encyclopedia_files/100px-Quadratic_Beziers_in_string_art.svg.png" width="100" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Quadratic_Beziers_in_string_art.svg/150px-Quadratic_Beziers_in_string_art.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Quadratic_Beziers_in_string_art.svg/200px-Quadratic_Beziers_in_string_art.svg.png 2x" data-file-width="512" data-file-height="1024"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Quadratic_Beziers_in_string_art.svg" class="internal" title="Enlarge"><img src="./Bézier curve - Wikipedia, the free encyclopedia_files/magnify-clip.png" width="15" height="11" alt=""></a></div>
Quadratic Béziers in string art: The end points (<b>•</b>) and control point (<b>×</b>) define the quadratic Bézier curve (<b>⋯</b>).</div>
</div>
</div>
<ul>
<li><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_surface" title="Bézier surface">Bézier surface</a></li>
<li><a href="http://en.wikipedia.org/wiki/Hermite_curve" title="Hermite curve" class="mw-redirect">Hermite curve</a></li>
<li><a href="http://en.wikipedia.org/wiki/NURBS" title="NURBS" class="mw-redirect">NURBS</a></li>
<li><a href="http://en.wikipedia.org/wiki/String_art" title="String art">String art</a> – Bézier curves are also formed by many common forms of string art, where strings are looped across a frame of nails.</li>
<li><a href="http://en.wikipedia.org/wiki/Variation_diminishing_property_of_B%C3%A9zier_curves" title="Variation diminishing property of Bézier curves" class="mw-redirect">Variation diminishing property of Bézier curves</a></li>
</ul>
<h2><span class="mw-headline" id="Notes">Notes</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=24" title="Edit section: Notes">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<div class="reflist columns references-column-count references-column-count-1" style="-moz-column-count: 1; -webkit-column-count: 1; column-count: 1; list-style-type: decimal;">
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-1"><span class="cite-accessibility-label">Jump up </span>^</a></b></span> <span class="reference-text">Image manipulation programs such as <a href="http://en.wikipedia.org/wiki/Inkscape" title="Inkscape">Inkscape</a>, <a href="http://en.wikipedia.org/wiki/Adobe_Photoshop" title="Adobe Photoshop">Adobe Photoshop</a>, and <a href="http://en.wikipedia.org/wiki/GIMP" title="GIMP">GIMP</a>.</span></li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-2"><span class="cite-accessibility-label">Jump up </span>^</a></b></span> <span class="reference-text">In animation applications such as <a href="http://en.wikipedia.org/wiki/Adobe_Flash" title="Adobe Flash">Adobe Flash</a>, <a href="http://en.wikipedia.org/wiki/Adobe_After_Effects" title="Adobe After Effects">Adobe After Effects</a>, <a href="http://en.wikipedia.org/wiki/Microsoft_Expression_Blend" title="Microsoft Expression Blend" class="mw-redirect">Microsoft Expression Blend</a>, <a href="http://en.wikipedia.org/wiki/Blender_(software)" title="Blender (software)">Blender</a>, <a href="http://en.wikipedia.org/wiki/Autodesk_Maya" title="Autodesk Maya">Autodesk Maya</a> and <a href="http://en.wikipedia.org/wiki/Autodesk_3ds_max" title="Autodesk 3ds max" class="mw-redirect">Autodesk 3ds max</a>.</span></li>
</ol>
</div>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=25" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ol class="references">
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-3"><span class="cite-accessibility-label">Jump up </span>^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.xlrotor.com/resources/files.shtml">http://www.xlrotor.com/resources/files.shtml</a></span></li>
<li id="cite_note-freetype-4"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-freetype_4-0"><span class="cite-accessibility-label">Jump up to: </span><sup><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-freetype_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.freetype.org/freetype2/docs/glyphs/glyphs-6.html">FreeType Glyph Conventions</a>, David Turner + Freetype Development Team, Freetype.org, retr May 2011</span></li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-5"><span class="cite-accessibility-label">Jump up </span>^</a></b></span> <span class="reference-text">John Burkardt. <a rel="nofollow" class="external text" href="http://people.sc.fsu.edu/~jburkardt/html/bezier_interpolation.html">"Forcing Bezier Interpolation"</a>.</span></li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-6"><span class="cite-accessibility-label">Jump up </span>^</a></b></span> <span class="reference-text"><span class="citation web">Shene, C.K. <a rel="nofollow" class="external text" href="http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/Bezier/de-casteljau.html">"Finding a Point on a Bézier Curve: De Casteljau's Algorithm"</a><span class="reference-accessdate">. Retrieved 6 September 2012</span>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AB%C3%A9zier+curve&rft.aufirst=C.K.&rft.aulast=Shene&rft.au=Shene%2C+C.K.&rft.btitle=Finding+a+Point+on+a+B%C3%A9zier+Curve%3A+De+Casteljau%27s+Algorithm&rft.genre=book&rft_id=http%3A%2F%2Fwww.cs.mtu.edu%2F~shene%2FCOURSES%2Fcs3621%2FNOTES%2Fspline%2FBezier%2Fde-casteljau.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span></li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-7"><span class="cite-accessibility-label">Jump up </span>^</a></b></span> <span class="reference-text"><span class="citation book">Farin, Gerald (1997). <i>Curves and surfaces for computer-aided geometric design</i> (4 ed.). <a href="http://en.wikipedia.org/wiki/Elsevier" title="Elsevier">Elsevier</a> Science & Technology Books. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-12-249054-5" title="Special:BookSources/978-0-12-249054-5">978-0-12-249054-5</a></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AB%C3%A9zier+curve&rft.au=Farin%2C+Gerald&rft.aufirst=Gerald&rft.aulast=Farin&rft.btitle=Curves+and+surfaces+for+computer-aided+geometric+design&rft.date=1997&rft.edition=4&rft.genre=book&rft.isbn=978-0-12-249054-5&rft.pub=Elsevier+Science+%26+Technology+Books&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span></li>
<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-8"><span class="cite-accessibility-label">Jump up </span>^</a></b></span> <span class="reference-text"><span class="citation book">Farin, Gerald (1997). <i>Curves and surfaces for computer-aided geometric design</i> (4 ed.). <a href="http://en.wikipedia.org/wiki/Elsevier" title="Elsevier">Elsevier</a> Science & Technology Books. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-12-249054-5" title="Special:BookSources/978-0-12-249054-5">978-0-12-249054-5</a></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AB%C3%A9zier+curve&rft.au=Farin%2C+Gerald&rft.aufirst=Gerald&rft.aulast=Farin&rft.btitle=Curves+and+surfaces+for+computer-aided+geometric+design&rft.date=1997&rft.edition=4&rft.genre=book&rft.isbn=978-0-12-249054-5&rft.pub=Elsevier+Science+%26+Technology+Books&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span></li>
<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve#cite_ref-9"><span class="cite-accessibility-label">Jump up </span>^</a></b></span> <span class="reference-text"><span class="citation web">Neil Dodgson (2000-09-25). <a rel="nofollow" class="external text" href="http://www.cl.cam.ac.uk/teaching/2000/AGraphHCI/SMEG/node5.html">"Some Mathematical Elements of Graphics: Rational B-splines"</a><span class="reference-accessdate">. Retrieved 2009-02-23</span>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AB%C3%A9zier+curve&rft.aulast=Neil+Dodgson&rft.au=Neil+Dodgson&rft.btitle=Some+Mathematical+Elements+of+Graphics%3A+Rational+B-splines&rft.date=2000-09-25&rft.genre=book&rft_id=http%3A%2F%2Fwww.cl.cam.ac.uk%2Fteaching%2F2000%2FAGraphHCI%2FSMEG%2Fnode5.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span></li>
</ol>
<ul>
<li>Rida T. Farouki, <i><a rel="nofollow" class="external text" href="http://mae.engr.ucdavis.edu/~farouki/bernstein.pdf">The Bernstein polynomial basis: A centennial retrospective</a></i>, Computer Aided Geometric Design, Volume 29, Issue 6, August 2012, Pages 379–419, <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="http://dx.doi.org/10.1016%2Fj.cagd.2012.03.001">10.1016/j.cagd.2012.03.001</a></li>
<li>Paul Bourke: <i>Bézier Surfaces (in 3D)</i>, <a rel="nofollow" class="external free" href="http://local.wasp.uwa.edu.au/~pbourke/geometry/bezier/index.html">http://local.wasp.uwa.edu.au/~pbourke/geometry/bezier/index.html</a></li>
<li><a href="http://en.wikipedia.org/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>: <i>Metafont: the Program</i>, Addison-Wesley 1986, pp. 123–131. Excellent discussion of implementation details; available for free as part of the TeX distribution.</li>
<li>Dr Thomas Sederberg, BYU <i>Bézier curves</i>, <a rel="nofollow" class="external free" href="http://www.tsplines.com/resources/class_notes/Bezier_curves.pdf">http://www.tsplines.com/resources/class_notes/Bezier_curves.pdf</a></li>
<li>J.D. Foley <i>et al.</i>: <i><a href="http://en.wikipedia.org/wiki/Computer_Graphics:_Principles_and_Practice#Second_Edition_in_C" title="Computer Graphics: Principles and Practice">Computer Graphics: Principles and Practice in C</a></i> (2nd ed., Addison Wesley, 1992)</li>
<li>Rajiv Chandel: "<a rel="nofollow" class="external text" href="http://codingg.blogspot.in/2014/03/implementing-bezier-curves-in-games.html">Implementing Bezier Curves in games</a>"</li>
</ul>
<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="http://en.wikipedia.org/w/index.php?title=B%C3%A9zier_curve&action=edit§ion=26" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul>
<li><a rel="nofollow" class="external text" href="http://www.ams.org/featurecolumn/archive/bezier.html#2">From Bézier to Bernstein</a> Feature Column from <a href="http://en.wikipedia.org/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a></li>
<li><span id="CITEREFHazewinkel2001" class="citation">Hazewinkel, Michiel, ed. (2001), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=p/b110460">"Bézier curve"</a>, <i><a href="http://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="http://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4" title="Special:BookSources/978-1-55608-010-4">978-1-55608-010-4</a></span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fen.wikipedia.org%3AB%C3%A9zier+curve&rft.atitle=Encyclopedia+of+Mathematics&rft.btitle=B%C3%A9zier+curve&rft.date=2001&rft.genre=bookitem&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dp%2Fb110460&rft.isbn=978-1-55608-010-4&rft.pub=Springer&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li>
<li><a rel="nofollow" class="external text" href="http://pomax.github.io/bezierinfo">A Primer on Bézier Curves</a> — A detailed explanation of implementing Bézier curves and associated graphics algorithms, with interactive graphics.</li>
<li><span class="citation mathworld" id="Reference-Mathworld-B.C3.A9zier_Curve"><a href="http://en.wikipedia.org/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, "<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BezierCurve.html">Bézier Curve</a>", <i><a href="http://en.wikipedia.org/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</span></li>
</ul>
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<li class="interlanguage-link interwiki-ar"><a href="http://ar.wikipedia.org/wiki/%D9%85%D9%86%D8%AD%D9%86%D9%89_%D8%A8%D9%8A%D8%B2%D9%8A%D9%87" title="منحنى بيزيه – Arabic" lang="ar" hreflang="ar">العربية</a></li>
<li class="interlanguage-link interwiki-bn"><a href="http://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%87%E0%A6%9C%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A7%87_%E0%A6%AC%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%B0%E0%A7%87%E0%A6%96%E0%A6%BE" title="বেজিয়ে বক্ররেখা – Bengali" lang="bn" hreflang="bn">বাংলা</a></li>
<li class="interlanguage-link interwiki-ca"><a href="http://ca.wikipedia.org/wiki/Corba_de_B%C3%A9zier" title="Corba de Bézier – Catalan" lang="ca" hreflang="ca">Català</a></li>
<li class="interlanguage-link interwiki-cs"><a href="http://cs.wikipedia.org/wiki/B%C3%A9zierova_k%C5%99ivka" title="Bézierova křivka – Czech" lang="cs" hreflang="cs">Čeština</a></li>
<li class="interlanguage-link interwiki-da"><a href="http://da.wikipedia.org/wiki/B%C3%A9zierkurve" title="Bézierkurve – Danish" lang="da" hreflang="da">Dansk</a></li>
<li class="interlanguage-link interwiki-de"><a href="http://de.wikipedia.org/wiki/B%C3%A9zierkurve" title="Bézierkurve – German" lang="de" hreflang="de">Deutsch</a></li>
<li class="interlanguage-link interwiki-el"><a href="http://el.wikipedia.org/wiki/%CE%9A%CE%B1%CE%BC%CF%80%CF%8D%CE%BB%CE%B7_%CE%9C%CF%80%CE%B5%CE%B6%CE%B9%CE%AD" title="Καμπύλη Μπεζιέ – Greek" lang="el" hreflang="el">Ελληνικά</a></li>
<li class="interlanguage-link interwiki-es"><a href="http://es.wikipedia.org/wiki/Curva_de_B%C3%A9zier" title="Curva de Bézier – Spanish" lang="es" hreflang="es">Español</a></li>
<li class="interlanguage-link interwiki-eo"><a href="http://eo.wikipedia.org/wiki/Kurbo_de_B%C3%A9zier" title="Kurbo de Bézier – Esperanto" lang="eo" hreflang="eo">Esperanto</a></li>
<li class="interlanguage-link interwiki-eu"><a href="http://eu.wikipedia.org/wiki/B%C3%A9zier_kurba" title="Bézier kurba – Basque" lang="eu" hreflang="eu">Euskara</a></li>
<li class="interlanguage-link interwiki-fr"><a href="http://fr.wikipedia.org/wiki/Courbe_de_B%C3%A9zier" title="Courbe de Bézier – French" lang="fr" hreflang="fr">Français</a></li>
<li class="interlanguage-link interwiki-ko"><a href="http://ko.wikipedia.org/wiki/%EB%B2%A0%EC%A7%80%EC%97%90_%EA%B3%A1%EC%84%A0" title="베지에 곡선 – Korean" lang="ko" hreflang="ko">한국어</a></li>
<li class="interlanguage-link interwiki-hr"><a href="http://hr.wikipedia.org/wiki/B%C3%A9zierova_krivulja" title="Bézierova krivulja – Croatian" lang="hr" hreflang="hr">Hrvatski</a></li>
<li class="interlanguage-link interwiki-id"><a href="http://id.wikipedia.org/wiki/Kurva_B%C3%A9zier" title="Kurva Bézier – Indonesian" lang="id" hreflang="id">Bahasa Indonesia</a></li>
<li class="interlanguage-link interwiki-it"><a href="http://it.wikipedia.org/wiki/Curva_di_B%C3%A9zier" title="Curva di Bézier – Italian" lang="it" hreflang="it">Italiano</a></li>
<li class="interlanguage-link interwiki-he"><a href="http://he.wikipedia.org/wiki/%D7%A2%D7%A7%D7%95%D7%9D_%D7%91%D7%96%D7%99%D7%99%D7%94" title="עקום בזייה – Hebrew" lang="he" hreflang="he">עברית</a></li>
<li class="interlanguage-link interwiki-lt"><a href="http://lt.wikipedia.org/wiki/Bezj%C4%97_kreiv%C4%97" title="Bezjė kreivė – Lithuanian" lang="lt" hreflang="lt">Lietuvių</a></li>
<li class="interlanguage-link interwiki-hu"><a href="http://hu.wikipedia.org/wiki/B%C3%A9zier_g%C3%B6rbe" title="Bézier görbe – Hungarian" lang="hu" hreflang="hu">Magyar</a></li>
<li class="interlanguage-link interwiki-mk FA" title="This is a featured article in this language."><a href="http://mk.wikipedia.org/wiki/%D0%91%D0%B5%D0%B7%D0%B8%D1%98%D0%B5%D0%BE%D0%B2%D0%B0_%D0%BA%D1%80%D0%B8%D0%B2%D0%B0" title="Безијеова крива – Macedonian" lang="mk" hreflang="mk">Македонски</a></li>
<li class="interlanguage-link interwiki-nl"><a href="http://nl.wikipedia.org/wiki/B%C3%A9zierkromme" title="Bézierkromme – Dutch" lang="nl" hreflang="nl">Nederlands</a></li>
<li class="interlanguage-link interwiki-ja"><a href="http://ja.wikipedia.org/wiki/%E3%83%99%E3%82%B8%E3%82%A7%E6%9B%B2%E7%B7%9A" title="ベジェ曲線 – Japanese" lang="ja" hreflang="ja">日本語</a></li>
<li class="interlanguage-link interwiki-no"><a href="http://no.wikipedia.org/wiki/B%C3%A9zier-kurve" title="Bézier-kurve – Norwegian (bokmål)" lang="no" hreflang="no">Norsk bokmål</a></li>
<li class="interlanguage-link interwiki-pl GA" title="This is a good article in this language."><a href="http://pl.wikipedia.org/wiki/Krzywa_B%C3%A9ziera" title="Krzywa Béziera – Polish" lang="pl" hreflang="pl">Polski</a></li>
<li class="interlanguage-link interwiki-pt"><a href="http://pt.wikipedia.org/wiki/Curva_de_B%C3%A9zier" title="Curva de Bézier – Portuguese" lang="pt" hreflang="pt">Português</a></li>
<li class="interlanguage-link interwiki-ro"><a href="http://ro.wikipedia.org/wiki/Curb%C4%83_B%C3%A9zier" title="Curbă Bézier – Romanian" lang="ro" hreflang="ro">Română</a></li>
<li class="interlanguage-link interwiki-ru"><a href="http://ru.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%B0%D1%8F_%D0%91%D0%B5%D0%B7%D1%8C%D0%B5" title="Кривая Безье – Russian" lang="ru" hreflang="ru">Русский</a></li>
<li class="interlanguage-link interwiki-sk"><a href="http://sk.wikipedia.org/wiki/B%C3%A9zierova_krivka" title="Bézierova krivka – Slovak" lang="sk" hreflang="sk">Slovenčina</a></li>
<li class="interlanguage-link interwiki-sl"><a href="http://sl.wikipedia.org/wiki/B%C3%A9zierova_krivulja" title="Bézierova krivulja – Slovenian" lang="sl" hreflang="sl">Slovenščina</a></li>
<li class="interlanguage-link interwiki-fi"><a href="http://fi.wikipedia.org/wiki/B%C3%A9zier-k%C3%A4yr%C3%A4" title="Bézier-käyrä – Finnish" lang="fi" hreflang="fi">Suomi</a></li>
<li class="interlanguage-link interwiki-sv"><a href="http://sv.wikipedia.org/wiki/B%C3%A9zier-kurva" title="Bézier-kurva – Swedish" lang="sv" hreflang="sv">Svenska</a></li>
<li class="interlanguage-link interwiki-th"><a href="http://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99%E0%B9%82%E0%B8%84%E0%B9%89%E0%B8%87%E0%B9%80%E0%B8%9A%E0%B8%8B%E0%B8%B4%E0%B9%80%E0%B8%A2%E0%B8%A3%E0%B9%8C" title="เส้นโค้งเบซิเยร์ – Thai" lang="th" hreflang="th">ไทย</a></li>
<li class="interlanguage-link interwiki-tr"><a href="http://tr.wikipedia.org/wiki/B%C3%A9zier_e%C4%9Frisi" title="Bézier eğrisi – Turkish" lang="tr" hreflang="tr">Türkçe</a></li>
<li class="interlanguage-link interwiki-uk"><a href="http://uk.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D0%B2%D0%B0_%D0%91%D0%B5%D0%B7%27%D1%94" title="Крива Без'є – Ukrainian" lang="uk" hreflang="uk">Українська</a></li>
<li class="interlanguage-link interwiki-vi"><a href="http://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_cong_B%C3%A9zier" title="Đường cong Bézier – Vietnamese" lang="vi" hreflang="vi">Tiếng Việt</a></li>
<li class="interlanguage-link interwiki-zh"><a href="http://zh.wikipedia.org/wiki/%E8%B2%9D%E8%8C%B2%E6%9B%B2%E7%B7%9A" title="貝茲曲線 – Chinese" lang="zh" hreflang="zh">中文</a></li>
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