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Bézier curve

Curve used in computer graphics and related fields / From Wikipedia, the free encyclopedia

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A Bézier curve (/ˈbɛz.i./ BEH-zee-ay)[1] is a parametric curve used in computer graphics and related fields.[2] A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. The Bézier curve is named after French engineer Pierre Bézier (1910–1999), who used it in the 1960s for designing curves for the bodywork of Renault cars.[3] Other uses include the design of computer fonts and animation.[3] Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces.[3] The Bézier triangle is a special case of the latter.

Cubic Bézier curve with four control points
The basis functions on the range t in [0,1] for cubic Bézier curves: blue: y = (1 − t)3, green: y = 3(1 − t)2t, red: y = 3(1 − t)t2, and cyan: y = t3.

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,[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 the time domain, particularly in animation,[4][note 2] user interface design and smoothing cursor trajectory in eye gaze controlled interfaces.[5] For example, 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 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.

This also applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear.