<<
i>graphics
i>> 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)