Bezier curve

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Bézier curve

[¦bāz·yā ′kərv]
(computer science)
A curve in a drawing program that is defined mathematically, and whose shape can be altered by dragging either of its two interior determining points with a mouse.
A simple smooth curve whose shape is determined by a mathematical formula from the locations of four points, the two end points of the curve and two interior points.

Bezier curve

A type of curve defined by mathematical formulae, used in computer graphics. A curve with coordinates P(u), where u varies from 0 at one end of the curve to 1 at the other, is defined by a set of n+1 "control points" (X(i), Y(i), Z(i)) for i = 0 to n.

P(u) = Sum i=0..n [(X(i), Y(i), Z(i)) * B(i, n, u)]

B(i, n, u) = C(n, i) * u^i * (1-u)^(n-i)

C(n, i) = n!/i!/(n-i)!

A Bezier curve (or surface) is defined by its control points, which makes it invariant under any affine mapping (translation, rotation, parallel projection), and thus even under a change in the axis system. You need only to transform the control points and then compute the new curve. The control polygon defined by the points is itself affine invariant.

Bezier curves also have the variation-diminishing property. This makes them easier to split compared to other types of curve such as Hermite or B-spline.

Other important properties are multiple values, global and local control, versatility, and order of continuity.

References in periodicals archive ?
Caption: Figure 1: Neutrosophic Bezier curves for data in Table 1.
Bezier curves are used to fit an existing data set while maintaining tangency and curvature conditions.
The proposed algorithm creates quadratic Bezier curves on some specific markups and interpolates the distances between them.
Recently, new functions are used to design the follower motion; for example, spline (Nguyen & Kim, 2007; Mermelstein & Acar, 2003), B-spline (Jiang & Iwai, 2009), and Bezier curves (Hidalgo et al.
This property makes the corresponding curves have variation diminishing property, which is one of the important properties of dominant Bezier curves and B-spline curves.
Their topics include affine and conformal transformations of rational Bezier curves, the G2-congruence classes of curves in the purely imaginary octonions, surfaces in the four-dimensional Euclidean and Minkowski space, radial transversal light-like hypersurfaces of almost complex manifolds with Norden metric, and the sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold.
If, like me, you associate Citroens with being all voluptuous swoops and extravagant Bezier curves, the rather cubist C4 Picasso might come as a bit of a surprise.
Because they are created using Bezier curves, not pixels, vector images can be enlarged substantially without compromising resolution.
Synthesis of displacement functions by Bezier curves in constant-breadth cams with parallel flat-faced double translating and oscillating followers, Mech.
They cover polynomials, Bezier curves, rational Bezier curves B-splines, approximation, spline curves, multivariate splines, surfaces and solids, and finite elements.
In this context piecewise cubics known as Bezier curves ([5]) have been studied extensively.
8] have evaluated the possibilities to utilize Bezier curves in different types of practical applications to improve the local information gained from the surface model.