Graphics Reference
In-Depth Information
Figure 11.10.
A Bézier curve.
curve at the end. The Bézier curve in Figure 11.10 shows some of basic properties
possessed by Bézier curves: it lies in the convex hull of the control points, it starts at
the first control point, ends at the last one, and is tangent to the first and last segment
of the characteristic polygon there.
Another important property of Bézier curves is that they are affinely invariant.
This follows both from Theorem 11.2.2.3 and from their recursive definition and the
fact that affine maps preserve barycentric coordinates (Theorem 2.5.9 in [AgoM05]).
As indicated earlier in Section 11.2.2, this is important for graphics because it says
that to compute a Bézier curve that has been moved by an affine map, such as a rota-
tion, all we have to do is compute the image of the control points and then recom-
pute the curve from those new points. Not all curves have this property.
Although it is usually convenient to restrict the domain of a Bézier curve to [0,1],
this is not necessary. More precisely, given any interval [a,b], a curve q : [a,b] Æ
R
m
of
the form
p
va
ba
-
-
Ê
Ë
ˆ
¯
,
()
=
qv
(11.60)
where p(u) is the Bézier curve on [0,1] defined by formula (11.51), is called a
Bézier
curve
. Such a curve satisfies all the properties that the original Bézier curve satisfied.
In particular, the important de Casteljau algorithm also applies to q(v) because the
algorithm really only uses barycentric coordinates and does not care about the end-
points of the domain. See the general de Casteljau algorithm in Section 11.5.2. All of
this is usually summarized by saying that Bézier curves are
invariant under affine
parameter transformations
.
In conclusion, the way that one works with Bézier curves in practice on an inter-
active graphics system is:
(1) One sketches a curve by hand.
(2) One specifies vertices that outline the shape.
(3) One then moves, adds, or deletes vertices as necessary to improve the shape
of the Bézier curve that is generated from these vertices.
The advantage of the Bézier curve over the Hermite curve is that it is more intu-
itive to specify four points than two points and two tangent vectors. Furthermore, the
fact that the curve lies in the convex hull of its control points makes clipping easier.
One first clips this convex hull against the window. If they do not intersect, then the
curve will be outside the window.
Two problems with the general Bézier curve are
