Bezier curves are used in computer graphics to produce curves which appear reasonably smooth at all scales . Mathematically, they are a special case of cubic Hermite interpolation. What this means is that curves are constructed as a sequence of cubic segments, rather than linear ones. But whereas Hermite interpolating polynomials are constructed in terms of derivatives at endpoints, Bezier curves use a construction due to Sergei Bernstein, in which the interpolating polynomials depend on certain control points. The mathematics of these curves is classical, but it was a French automobile engineer Pierre Bezier who introduced their use in computer graphics.
One reason these curves are used so much in computer graphics is that they are very efficient to construct, since a simple recursion process means that the basic arithmetic operation needed to build the points along one is just division by two. For this reason, also, the most efficient implementations use scaled integers instead of floating point numbers as basic numerical data.
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.
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