Graphics Reference
In-Depth Information
t
)
P
(1) +
higher order terms of t
P
(
t
)=
P
(1)
−
(1
−
=
V
p
{
1
−
p
(1
−
t
)
}
+
p
(1
−
t
)
V
p−
1
.
We observe that as
t
→
0, the Bezier polynomial lies on the line joining
V
0
and
V
1
, and for
t
1 on the line joining
V
p−
1
and
V
p
. This concludes that these
lines are tangents to the curve P(t) at
V
0
and
V
p
. one can choose, therefore,
the end control points in such a way that that they lie on a straight line.
Hence, two pieces of curves can be easily drawn to maintain continuity at
their joining point, and as a result, this provides effectively a single spline
curve. For the B-B basis function in the model, the spline curve so obtained
is known as B-B spline curve and the underlying spline function is known as
the B-B spline or simply the Bezier spline.
→
p
Since
φ
ip
(
t
) = 1, the Bezier curve lies inside the convex hull defined
i
=0
by the control points. For cubic Bezier curve,
p
= 3. The control polygon
corresponding to
p
= 3 consists of four control vertices, namely,
V
0
,V
1
,V
2
,V
3
,
and the Bezier curve is
t
)
3
V
0
+3
t
(1
t
)
2
V
1
+3
t
2
(1
t
)
V
2
+
t
3
V
3
.
P
(
t
)=(1
−
−
−
(1.16)
The Bernstein basis functions in this case are as follows:
φ
03
(
t
)=1
t
3
=1
3
t
2
+3
t
t
3
−
−
−
t
)
2
=3
t
6
t
2
+3
t
3
φ
13
(
t
)=3
t
(1
−
−
φ
23
(
t
)=3
t
2
(1
t
)=3
t
2
3
t
3
−
−
φ
33
(
t
)=
t
3
.
Though the cubic Bezier curve is widely used in computer graphics [133],
one can use, as well, its quadratic version to speed up the procedure, without
degrading the quality of drawing. For a quadratic Bezier curve,
p
=2andthe
control polygon consists of three points. The Bernstein basis in this case are
φ
02
(
t
)=(1
t
)
2
=1
2
t
+
t
2
−
−
2
t
2
φ
12
(
t
) = 2(1
−
t
)
t
=2
t
−
φ
22
(
t
)=
t
2
.
In the polynomial form, the Bezier curve is
P
(
t
)=
t
2
(
V
0
+
V
2
−
2
V
1
)+
t
(2
V
1
−
2
V
0
)+
V
0
.
(1.17)
This is a second degree polynomial and can be computed much faster than in
Horner's process [133].
One should note that for a cubic Bezier curve, the basis function
φ
13
attains its maximum at
t
=
3
and the maximum value is
φ
13
(
1
3
)=
4
9
,
(1.18)
while
φ
23
has the maximum at
t
=
3
with
φ
23
(
2
3
)=
4
9
.
(1.19)
Search WWH ::
Custom Search