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 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.

[What do these properties mean?]

(1996-06-12)