In the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.
Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies. The curves were first developed in 1959 by Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bézier curves.
In vector graphics, Bézier curves are an important tool used to model smooth curves that can be scaled indefinitely. "Paths," as they are commonly referred to in image manipulation programs^{[note 1]} are combinations of linked Bézier curves. Paths are not bound by the limits of rasterized images and are intuitive to modify. Bézier curves are also used in animation as a tool to control motion.^{[note 2]}
Bézier curves are also commonly used over the time domain, particularly in animation and interface design. Thus, a Bézier curve is often used to describe or control the velocity over time of an object moving from A to B. For example, an icon might "easeinout" or follow a "cubic Bézier" in moving from A to B, rather than simply moving at a fixed number of pixels per step. Indeed, when animators or interface designers discuss the "physics" or "feel" of an operation, they often are referring to the particular Bézier curve used to control the velocity over time of the move in question.
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Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively. Affine transformations such as translation, scaling and rotation can be applied on the curve by applying the respective transform on the control points of the curve.
Quadratic and cubic Bézier curves are most common; higher degree curves are more expensive to evaluate. When more complex shapes are needed, low order Bézier curves are patched together. This is commonly referred to as a "path" in programs like Adobe Illustrator or Inkscape. These polyBézier curves can also be seen in the SVG file format. To guarantee smoothness, the control point at which two curves meet must be on the line between the two control points on either side.
The simplest method for scan converting (rasterizing) 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 ≤ t ≤ 0.5 and 0.5 ≤ t ≤ 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 spline 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.
In animation applications, such as Adobe Flash and Synfig, or in applications like Game Maker, 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.
Given points P_{0} and P_{1}, a linear Bézier curve is simply a straight line between those two points. The curve is given by
and is equivalent to linear interpolation.
A quadratic Bézier curve is the path traced by the function B(t), given points P_{0}, P_{1}, and P_{2},
A quadratic Bézier curve is also a parabolic segment.
TrueType fonts use Bézier splines composed of quadratic Bézier curves.
Four points P_{0}, P_{1}, P_{2} and P_{3} in the plane or in threedimensional space define a cubic Bézier curve. The curve starts at P_{0} going toward P_{1} and arrives at P_{3} coming from the direction of P_{2}. Usually, it will not pass through P_{1} or P_{2}; these points are only there to provide directional information. The distance between P_{0} and P_{1} determines "how long" the curve moves into direction P_{2} before turning towards P_{3}.
The parametric form of the curve is:
Since the lines and are the tangents of the Bézier curve at and , respectively, cubic Bézier interpolation is essentially the same as cubic Hermite interpolation.
Modern imaging systems like PostScript, Asymptote and Metafont use Bézier splines composed of cubic Bézier curves for drawing curved shapes.
The Bézier curve of degree n can be generalized as follows. Given points P_{0}, P_{1},..., P_{n}, the Bézier curve is
where is the binomial coefficient.
For example, for n = 5:
This formula can be expressed recursively as follows: Let denote the Bézier curve determined by the points P_{0}, P_{1},..., P_{n}. Then
In other words, the degree n Bézier curve is a linear interpolation between two degree n − 1 Bézier curves.
Some terminology is associated with these parametric curves. We have
where the polynomials
are known as Bernstein basis polynomials of degree n, defining t^{0} = 1 and (1  t)^{0} = 1. The binomial coefficient, , has the alternative notation,
The points P_{i} are called control points for the Bézier curve. The polygon formed by connecting the Bézier points with lines, starting with P_{0} and finishing with P_{n}, is called the Bézier polygon (or control polygon). The convex hull of the Bézier polygon contains the Bézier curve.
Animation of a linear Bézier curve, t in [0,1] 
The t in the function for a linear Bézier curve can be thought of as describing how far B(t) is from P_{0} to P_{1}. For example when t=0.25, B(t) is one quarter of the way from point P_{0} to P_{1}. As t varies from 0 to 1, B(t) describes a curved line from P_{0} to P_{1}.
For quadratic Bézier curves one can construct intermediate points Q_{0} and Q_{1} such that as t varies from 0 to 1:
Construction of a quadratic Bézier curve  Animation of a quadratic Bézier curve, t in [0,1] 
For higherorder curves one needs correspondingly more intermediate points. For cubic curves one can construct intermediate points Q_{0}, Q_{1} & Q_{2} that describe linear Bézier curves, and points R_{0} & R_{1} that describe quadratic Bézier curves:
Construction of a cubic Bézier curve  Animation of a cubic Bézier curve, t in [0,1] 
For fourthorder curves one can construct intermediate points Q_{0}, Q_{1}, Q_{2} & Q_{3} that describe linear Bézier curves, points R_{0}, R_{1} & R_{2} that describe quadratic Bézier curves, and points S_{0} & S_{1} that describe cubic Bézier curves:
Construction of a quartic Bézier curve  Animation of a quartic Bézier curve, t in [0,1] 
(See also a construction of a fifthorder Bézier curve.)
Sometimes it is desirable to express the Bézier curve as a polynomial instead of a sum of less straightforward Bernstein polynomials. Application of the binomial theorem to the definition of the curve followed by some rearrangement will yield:
where
This could be practical if can be computed prior to many evaluations of ; however one should use caution as high order curves may lack numeric stability (de Casteljau's algorithm should be used if this occurs). Note that the product of no numbers is 1.
The rational Bézier curve adds adjustable weights to provide closer approximations to arbitrary shapes. The numerator is a weighted Bernsteinform Bézier curve and the denominator is a weighted sum of Bernstein polynomials. Rational Bézier curves can, among other uses, be used to represent segments of conic sections exactly.^{[1]}
Given n + 1 control points P_{i}, the rational Bézier curve can be described by:
or simply
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Popularized by Pierre Bézier.
Singular 
Plural 
Bézier curve (plural Bézier curves)

