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 ?
They studied geometric models of fuzzy and intuitionistic fuzzy data and gave fuzzy interpolation and Bezier curve modeling.
In the present study, a third order Bezier curve is used to describe the geometry.
The Bezier curve are adjusted based on the previous results.
Google earth and GPS were used to generate the target path using the Bezier curve method and optimized route is chosen.
However, the Bezier curve is a single curve segment.
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.
3] derived sharp bounds for the distance between a planar parametric Bezier curve and parameterizations of its control polyhedron based on the Greville abscissae.
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.
Enhanced Bezier curve models incorporating local information, 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol.
Then, the quadratic Bezier Curve is built, which will pass through pointp3v at axis [t.
2005) designed streamlined dies using Bezier curve and polynomial equations.