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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)
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