Graphics Reference
In-Depth Information
P
1
P
3
P
0
P
2
FIGURE B.44
Cubic Bezier curve segment.
FIGURE B.45
Composite cubic Bezier curve showing tangents and colinear control points.
Continuity between adjacent Bezier segments can be controlled by colinearity of the control points
on either side of the shared beginning/ending point of the two curve segments where they join
(
Figure B.45
). In addition, the Bezier curve form allows one to define a curve of arbitrary order. If
three interior control points are used, then the resulting curve will be quartic; if four interior control
discussion.
B.5.10
De Casteljau construction of Bezier curves
The de Casteljau method is a way to geometrically construct a Bezier curve.
Figure B.46
shows the
construction of a point at
u ¼
1/3. This method constructs a point
u
along the way between paired con-
trol points (identified by a “1” in
Figure B.46
). Then points are constructed
u
along the way between
points just previously constructed. These new points are marked “2” in
Figure B.46
.
In the cubic case,
in which there were four initial points, there are two newly constructed points. The point on the curve is
constructed by going
u
along the way between these two points. This can be done for any values of
u
and for any order of curve. Higher order Bezier curves require more iterations to produce the final point
on the curve.
B.5.11
Tension, continuity, and bias control
Often an animator wants better control over the interpolation of key frames than the standard interpo-
lating splines provide. For better control of the shape of an interpolating curve, Kochanek [
11
] suggests
a parameterization of the internal tangent vectors based on the three values: tension, continuity, and
bias. The three parameters are explained by decomposing each internal tangent vector into an incoming
part and an outgoing part. These tangents are referred to as the left and right parts, respectively, and are
notated by
T
i
and
T
i
for the tangents at
P
i
.
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