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 were widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies.

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.

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.

The points P < sub > i </ sub > are called control points for the Bézier curve.

At the high-end, where one is willing to trade-off higher rendering times for increased image quality and precision, smooth surface representations such as Bézier patches, NURBS or Subdivision surfaces are used.

The edges of the triangle are themselves Bézier curves, with the same control points as the Bézier triangle.

An advantage of Bézier triangles in computer graphics is, they are smooth, and can easily be approximated by regular triangles, by recursively dividing the Bézier triangle into two separate Bézier triangles, until they are considered sufficiently small, using only addition and division by two, not requiring any floating point arithmetic whatsoever.

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.

A quadratic Bézier curve is also a parabolic segment.

* A curve can be split at any point into two subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

De Casteljau's algorithm can also be used to split a single Bézier curve into two Bézier curves at an arbitrary parameter value.

* Outline fonts ( also called vector fonts ) use Bézier curves, drawing instructions and mathematical formulae to describe each glyph, which make the character outlines scalable to any size.

A Bézier curve is a parametric curve frequently used in computer graphics and related fields.

When animators or interface 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.

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.

* Patch, a 3-D Bézier curve used in computer graphics, or a primitive in some 3-D software packages

Further command letters ( C, S, Q, T and A ) precede data that is used to draw various Bézier and elliptical curves.

Usually cubic Bézier curves are used, and additional control points ( called handles ) are added to define the shape of each curve.

Bézier surfaces are a species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling.

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

But because Bézier published the results of his work, the average computer graphics user today recognizes splines — which are represented with control points lying off the curve itself — as Bézier splines, while de Casteljau ’ s name is only known and used for the algorithms he developed to evaluate parametric surfaces.

* Paths: made with the Pencil tool, which allows freehand drawing of paths ; the Pen tool, which allows the user to create a Bézier spline node-by-node curves and lines in the same path ; the Calligraphy tool, which can be used to draw freehand calligraphic or brush-like strokes, or the Paint Bucket tool, which fills bounded areas of the image.

The most commonly used splines are cubic spline, i. e., of order 3 — in particular, cubic B-spline and cubic Bézier spline.

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.

Quaternion Slerps are commonly used to construct smooth animation curves by mimicking affine constructions like the de Casteljau algorithm for Bézier curves.

Depending on the type of user interface, editing can be realized via an element ’ s control points, which are most obvious and common for Bézier curves, or via higher level tools such as spline modeling or hierarchical editing.

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.

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