Graphics Reference
In-Depth Information
tion. Their use in computer graphics is well known, while the use in image data
approximation for image compression or feature extraction is challenging. We
shall discuss the eciency of the polynomial in each area. Before doing that,
we shall elaborate on Bezier curves.
1.4.1 Bezier-Bernstein Curves
This class of curves was first proposed by Bezier [22, 17]. The parametric form
of the curves is
X = P x ( t )
(1.11)
Y = P y ( t ) .
(1.12)
Let ( x 0 ,y 0 ) , ( x 1 ,y 1 )
( x p ,y p )be( p + 1) ordered points in a plane. The
Bezier curve associated with the polygon through the aforementioned points
is the vector valued Bernstein polynomial and is given by
···
p
P x ( t )=
φ ip ( t ) x i
(1.13)
i =0
p
P y ( t )=
φ ip ( t ) y i
(1.14)
i =0
where φ ip ( t ) s 's are the binomial probability density function of (1.2). In the
vector form, equation (1.13) and equation (1.14) can be written as
p
P ( t )=
φ ip ( t ) V i .
(1.15)
i =0
The points V 0 ,V 1 ,
,V p are known as the guiding points or the control points
for the curve P(t). From equation (1.15) it is seen that
···
P (0) = V 0 and P (1) = V p .
Thus, the average of t significantly extends from 0 to 1. The derivative of P ( t )
is
p
i
p
1
t ) p− 1 v 0 +
t ) p−i
P ( t )=
it i− 1 (1
p (1
{
i =1
t ) p−i− 1
i ) t i (1
V i + pt p− 1 v p .
( p
}
Now P (0) = p ( V 1
V 0 )and P (1) = p ( V p
V p− 1 ). Thus the Taylor series
expansion near zero is
P ( t )= P (0) + tP (0) + higher order terms of t
= V 0 (1
pt )+
···
and an expansion near one is
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