Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.

In vector graphics, Bézier curves are used to model smooth curves that can be scaled indefinitely.

" Paths ," as they are commonly referred to in image manipulation programs, are combinations of linked Bézier curves.

Bézier curves are also used in animation as a tool to control motion.

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.

But the study of these curves was first developed in 1959 by mathematician Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bézier curves.

Bézier curves are widely used in computer graphics to model smooth curves.

Quadratic and cubic Bézier curves are most common ; higher degree curves are more computationally expensive to evaluate.

When more complex shapes are needed, low order Bézier curves are patched together.

In animation applications, such as Adobe Flash and Synfig, 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.

TrueType fonts use Bézier splines composed of quadratic Bézier curves.

Modern imaging systems like PostScript, Asymptote, Metafont, and SVG use Bézier splines composed of cubic Bézier curves for drawing curved shapes.

Notable company projects were at GM ( Dr. Patrick J. Hanratty ) with DAC-1 ( Design Augmented by Computer ) 1964 ; Lockheed projects ; Bell GRAPHIC 1 and at Renault ( Bézier ) – UNISURF 1971 car body design and tooling.

Bézier surfaces were first described in 1962 by the French engineer Pierre Bézier who used them to design automobile bodies.

The pioneers of this development were Pierre Bézier who worked as an engineer at Renault, and Paul de Casteljau who worked at Citroën, both in France.

Adobe Illustrator, a vector drawing program based on the Bézier curve introduced in 1987 and Adobe Photoshop, written by brothers Thomas and John Knoll in 1990 were developed for use on MacIntosh computers.

Note that the intermediate points that were constructed are in fact the control points for two new Bézier curves, both exactly coincident with the old one.

OmniGraffle 5 added Bézier connecting lines ; previously connecting lines were bent to fit between their points which made them relatively easy to create, but difficult to control with any great precision.

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.

Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves ( or surfaces ) into NURBS, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition.

A Bézier curve is defined by a set of control points P < sub > 0 </ sub > through P < sub > n </ sub >, where n is called its order ( n

A quadratic Bézier curve is the path traced by the function B ( t ), given points P < sub > 0 </ sub >, P < sub > 1 </ sub >, and P < sub > 2 </ sub >,

Writing B < sub > P < sub > i </ sub >, P < sub > j </ sub >, P < sub > k </ sub ></ sub >( t ) for the quadratic Bézier curve defined by points P < sub > i </ sub >, P < sub > j </ sub >, and P < sub > k </ sub >, the cubic Bézier curve can be defined as a linear combination of two quadratic Bézier curves:

Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.

The polygon formed by connecting the Bézier points with lines, starting with P < sub > 0 </ sub > and finishing with P < sub > n </ sub >, is called the Bézier polygon ( or control polygon ).

* Some curves that seem simple, such as the circle, cannot be described exactly by a Bézier or piecewise Bézier curve ; though a four-piece cubic Bézier curve can approximate a circle ( see Bézier spline ), with a maximum radial error of less than one part in a thousand, when each inner control point ( or offline point ) is the distance horizontally or vertically from an outer control point on a unit circle.

* The curve at a fixed offset from a given Bézier curve, often called an offset curve ( lying " parallel " to the original curve, like the offset between rails in a railroad track ), cannot be exactly formed by a Bézier curve ( except in some trivial cases ).

However, probably the most important work on polynomial curves and sculptured surface was done by Pierre Bézier ( Renault ), Paul de Casteljau ( Citroen ), Steven Anson Coons ( MIT, Ford ), James Ferguson ( Boeing ), Carl de Boor ( GM ), Birkhoff ( GM ) and Garibedian ( GM ) in the 1960s and W. Gordon ( GM ) and R. Riesenfeld in the 1970s.

Some computer graphic systems make use of Bézier splines, which allow a curve to be bent in real time on a display screen to follow a set of coordinates, much in the way a French curve would be placed on a set of three or four points on paper.

* Pierre Bézier, French engineer and creator of Bézier curves

It was developed by French engineer Pierre Bézier for Renault in 1968.

* Pierre Bézier, an engineer and mathematician known for his work with Bézier curves

In 1959, while working at Citroën, he developed an algorithm for evaluating calculations on a certain family of curves, which would later be formalized and popularized by engineer Pierre Bézier, and the curves called Bézier curves.

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* Pierre Bézier ( Paris, 1927 ), inventor of computer-aided design