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| {{Contradict|article|"Definition"|about=the subject|date=August 2013}}
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| {{Formula missing descriptions|date=August 2010}}
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| In the [[mathematics|mathematical]] field of [[numerical analysis]] and in [[computer graphics]], a '''Bézier spline''' is a [[spline curve]] where each polynomial of the spline is in [[Bézier form]].
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| In other words,{{Contradiction-inline|reason=not other words, something very different, see Talk "Definition".|date=August 2013}} a '''Bézier spline''' is simply a series of [[Bézier curve]]s joined end to end where the last point of one curve coincides with the starting point of the next curve. Usually cubic Bézier curves are used, and additional control points (called handles) are added to define the shape of each curve.
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| A Bézier spline is similar to a [[polyline]] in that it connects a series of points, but whereas in polylines the points are connected by straight lines, in a Bézier spline the points are connected by Bézier curves.
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| [[Image:Beziergon.svg|thumb|right|200px|Beziergon - The red beziergon passes through the blue vertices, the green points are control points that determine the shape of the connecting Bézier curves]]
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| In [[geometry]], a '''beziergon''' (also called '''bezigon''' or '''polybezier''') is a closed path composed of [[Bezier curves|Bézier curves]]. It is similar to a [[polygon]] in that it connects a set of [[vertex (geometry)|vertices]] by lines, but whereas in polygons the vertices are connected by straight lines, in a beziergon the vertices are connected by Bézier curves.
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| <ref>[http://msdn2.microsoft.com/en-us/library/ms534244.aspx Microsoft polybezier API]
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| </ref><ref>[http://libpapyrus.sourceforge.net/guide_beziergon.html Papyrus beziergon API reference]
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| </ref><ref>[http://books.google.com/books?id=nFAEAAAAMBAJ&pg=PA85&lpg=PA85&dq=bezigon+curve&source=bl&ots=NgfxXCVYQq&sig=RVvSI3SH2j6LXNBh_Y10KjCLlEs&hl=en&sa=X&ei=XygXUfnWI6jm2gX36ICYBA&ved=0CC4Q6AEwADgU#v=onepage&q=bezigon%20curve&f=false "A better box of crayons"]. | |
| InfoWorld.
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| 1991.</ref>
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| Perhaps the most common use of Bézier splines is to describe the outline of each letter in a [[PostScript]] or [[PDF]] file.
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| Such outlines are composed of one beziergon for [[typeface anatomy|open letters]],
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| or multiple beziergons for closed letters. | |
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| Modern [[vector graphics]] and [[computer font]] systems like [[PostScript]], [[Asymptote (vector graphics language)|Asymptote]], [[Metafont]], [[OpenType]], and [[SVG]] use Bézier splines composed of '''cubic''' Bézier curves (3rd order curves) for drawing curved shapes.
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| [[TrueType]] fonts use Bézier splines composed of '''quadratic''' Bézier curves (2nd order curves) -- little pieces of parabolas.<ref name="lancaster">
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| [http://www.tinaja.com/glib/hack62.pdf Don Lancaster]
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| </ref> | |
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| A few early digital fonts use [[polygonal chain]]s—each [[letter form]] is composed of what are effectively 1st order Bézier curves—straight lines. | |
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| To describe a typical [[type design]] as a [[computer font]] to any given accuracy,
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| 3rd order splines (a series of 3rd order curves) require less data than 2nd order splines; and
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| 2nd order splines in turn require less data than a series of straight lines.<ref name="lancaster" />
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| This is true even though any one straight line segment requires less data than any one segment of a parabola; and that parabolic segment in turn requires less data than any one segment of a 3rd order curve. | |
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| ==Definition==
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| [[File:Sinc Function Approximation with Bezier Splines.svg|thumb|[[Sinc]] function approximated using Bézier splines ]]
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| Given a spline ''S'' of degree ''n'' with ''k'' knots ''x''<sub>''i''</sub> we can write the spline as a '''Bézier spline''' as:{{citation needed|date=December 2013}}
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| <math>
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| S(x) := \left\{
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| \begin{matrix}
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| S_0(x) := & \sum_{\nu=0}^{n} \beta_{\nu,0} b_{\nu,n}(x) & x \in [x_0, x_1] \\
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| S_1(x) := & \sum_{\nu=0}^{n} \beta_{\nu,1} b_{\nu,n}(x - x_1) & x \in [x_1, x_2] \\
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| \vdots & \vdots \\
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| S_{k-2}(x) := & \sum_{\nu=0}^{n} \beta_{\nu,k-2} b_{\nu,n}(x - x_{k -2}) & x \in [x_{k-2}, x_{k-1}] \\
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| \end{matrix}\right.
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| </math>
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| where the <math>b_{\nu,n}(x)</math> are the [[Bernstein polynomial#Definition|Bernstein basis polynomials]] of degree <math>n</math> and the <math>\beta_{\nu,i}</math> are the (unique) coefficients according to their [[linear combination]] for <math>S_i</math>.
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| ==Approximating circular arcs==
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| In case circular arc primitives are not supported in a particular environment, they may be approximated by [[Bézier curve]]s.<ref>{{Cite web
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| | last = Stanislav
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| | first = G. Adam
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| | title = Drawing a circle with Bézier Curves
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| | url=http://whizkidtech.redprince.net/bezier/circle/
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| | accessdate = 10 April 2010 }}
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| </ref> Commonly, four cubic segments are used to approximate a circle. It is desirable to find the length <math>\mathbf{k}</math> of control points which result in the least approximation error.
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| ===Using four curves===
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| Considering only the 90-degree [[unit circle|unit-circular]] arc in the [[Cartesian coordinate system#Quadrants and octants|first quadrant]], we define the endpoints <math>\mathbf{A}</math> and <math>\mathbf{B}</math> with control points <math>\mathbf{A'}</math> and <math>\mathbf{B'}</math>, respectively, as:
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| :<math>
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| \begin{align}
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| \mathbf{A} & = [0, 1] \\
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| \mathbf{A'} & = [\mathbf{k}, 1] \\
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| \mathbf{B'} & = [1, \mathbf{k}] \\
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| \mathbf{B} & = [1, 0] \\
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| \end{align}
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| </math>
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| From the definition of the cubic Bézier curve, we have:
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| :<math>\mathbf{C}(t)=(1-t)^3\mathbf{A} + 3(1-t)^2t\mathbf{A'}+3(1-t)t^2\mathbf{B'}+t^3\mathbf{B}</math>
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| With the point <math>\mathbf{C}(t=0.5)</math> as the midpoint of the arc, we may write the following two equations:
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| :<math>
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| \begin{align}
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| \mathbf{C} &= \frac{1}{8}\mathbf{A} + \frac{3}{8}\mathbf{A'}+\frac{3}{8}\mathbf{B'}+\frac{1}{8}\mathbf{B} \\
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| \mathbf{C} &= \sqrt{1/2} = \sqrt{2}/2
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| \end{align}
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| </math>
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| Solving these equations for the x-coordinate (and identically for the y-coordinate) yields:
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| :<math>\frac{0}{8}\mathbf + \frac{3}{8}\mathbf{k}+\frac{3}{8} + \frac{1}{8} = \sqrt{2}/2</math>
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| :<math>\mathbf{k} = \frac{4}{3}(\sqrt{2} - 1) \approx 0.5522847498</math>
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| ===General case===
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| We may compose a circle of radius <math>R</math> from an arbitrary number of cubic Bézier curves.<ref>{{Cite journal
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| | last = Riškus
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| | first = Aleksas
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| |date=October 2006
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| | title = APPROXIMATION OF A CUBIC BEZIER CURVE BY CIRCULAR ARCS AND VICE VERSA
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| | journal = INFORMATION TECHNOLOGY AND CONTROL
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| | volume = 35
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| | issue = 4
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| | pages = 371–378
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| | location = Department of Multimedia Engineering, Kaunas University of Technology
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| | issn = 1392-124X
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| | url = http://itc.ktu.lt/itc354/Riskus354.pdf}}</ref>
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| Let the arc start at point <math>\mathbf{A}</math> and end at point <math>\mathbf{B}</math>, placed at equal distances above and below the x-axis, spanning an arc of angle <math>\theta = 2\phi</math>:
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| :<math>\begin{align}
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| \mathbf{A}_x &= R\cos(\phi) \\
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| \mathbf{A}_y &= R\sin(\phi) \\
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| \mathbf{B}_x &= \mathbf{A}_x \\
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| \mathbf{B}_y &= -\mathbf{A}_y
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| \end{align}</math>
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| The control points may be written as:<ref>
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| {{Cite web
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| | last = DeVeneza
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| | first = Richard
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| | title = Drawing a circle with Bézier Curves
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| | url=http://www.tinaja.com/glib/bezcirc2.pdf
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| | accessdate = 10 April 2010 }}</ref>
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| :<math>\begin{align}
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| \mathbf{A'}_x &= \frac{4R - \mathbf{A}_x}{3} \\
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| \mathbf{A'}_y &= \frac{(R - \mathbf{A}_x)(3R - \mathbf{A}_x)}{3\mathbf{A}_y} \\
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| \mathbf{B'}_x &= \mathbf{A'}_x \\
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| \mathbf{B'}_y &= -\mathbf{A'}_y
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| \end{align}</math>
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| ===Examples===
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| <gallery widths="256px" heights="256px"> | |
| File:Circle and quadratic bezier.svg|Eight-segment quadratic Bézier spline (red) approximating a circle (black) with control points
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| File:Circle and cubic bezier.svg|Four-segment cubic Bézier spline (red) approximating a circle (black) with control points
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| </gallery>
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| ==References==
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| {{Reflist}}
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| {{Use dmy dates|date=July 2011}}
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| * {{cite book|last1=Koehler |last2=Dr. Ralph|title=2D/3D Graphics and Splines with Source Code|isbn= 0-7596-1187-4}}
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| {{DEFAULTSORT:Bezier Spline}}
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| [[Category:Splines]]
| |
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