Bezier curve

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 ?
With the development of CAD technology and modern processing technology, further research are taken place by using high order polynomial curve bezier curve B-spline curve etc.
In addition, the tangents of a point on a Bezier curve can be modified, allowing more precise tangent editing.
The bezier curve will deform automatically with the motion of her mouth as she speaks without further intervention, allowing the change of the color of her lips which can be graded to perfection.
A Bezier curve described by n + 1 control points (a polynomial curve of degree n or order n + 1) is shown in Equation 1 (Piegl and Tiller 1997, pp.
2005) designed streamlined dies using Bezier curve and polynomial equations.
In Section 4, we present an example from [2] involving the fitting of a Bezier curve to ordered two-dimensional data where all of the mixed partial derivatives are identically zero.
The new Bezier benchmark interpolates a set of points defined by the four points of a Bezier curve and stresses the ability of embedded microprocessors to perform division, multiplication, and scalar processing tasks.
Consider that the paintings that first garnered Murakami acclaim are composed--that is, digitally drawn--with Bezier curve programs.
The author is currently developing a 3D Bezier curve implementation for creases.
Thus the r-th derivative of a Bezier curve at an endpoint depends only on the r + 1 Bezier coefficients near (and including) that endpoint.
This paper presents the development of graphics using Bezier curve, but what is also suggested is the use of various functions arising as original language.
Let P(t) = (x(t), y(t)) (with t [member of] [0, 1]) be a rational Bezier curve of degree 5 where