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. (mathematics) 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

(graphics)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 theother, 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 evenunder a change in the axis system. You need only to transformthe control points and then compute the new curve. Thecontrol polygon defined by the points is itself affineinvariant.

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

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