Graphics Reference
In-Depth Information
φ
ip
is the ith basis function of order p.
V
i
,for
i
=0
,
1
,
p
defines a polygon
known as the Bezier control polygon. Bezier based his approximation method
on the classical
Bernstein polynomial approximation
. The Bernstein polyno-
mial approximation of degree p to an arbitrary real valued function
f
(
t
)is
···
p
f
(
i
B
ip
[
f
(
t
)] =
p
)
φ
ip
(
t
)
0
≤
t
≤
1
.
i
=0
Bezier's approach, therefore, specifies a well-ordered set of points, say p+1 in
number to do the approximation. These points
define a p-
sided polygon that is well suited to the problem of interactive design of smooth
free-form curves. Changing the values of
V
i
changes the polygon and hence,
changes the shape of the curve. Thus, the shape of the curve is controlled
through the shape of the polygon. In two dimensions, B-B polynomial repre-
sents a surface patch or a piece of a surface. The free-form drawing of curves
and surfaces is very useful in computer graphics. The ordered representative
points
f
(
i/p
) in equation (1.1) in the approximation mode are, therefore, the
guiding or control points in the design mode for curves.
{
V
i
,i
=0
,
1
,
···
p
}
Some Properties
One dimensional Bezier-Bernstein polynomial represents a curve that can be
generated from a set of ordered representative points, called the control points
or the guiding points. The line joining these control points is called the control
line of the polynomial. It reflects the shape of the curve that one wants to
draw or generate. Such curves have the following attractive properties:
•
They always interpolate the end control points, and the line joining two
consecutive points at either end is a tangent to the curve at that end point.
•
They remain always enclosed within the convex hull defined by the control
points.
•
They have the variation diminishing property, i.e., they do not exhibit any
oscillating behavior about any line more often than a sequence of lines
joining the control points.
•
They have the axis independence property, i.e., the drawing of the curve
does not depend on any axis.
•
They are ane invariant.
•
Determination of the polynomial order in drawing a curve is easy and
straightforward. It is always one less than the number of vertices of the
control polygon.
1.4 Use in Computer Graphics and Image Data
Approximation
Due to the attractive properties of the Bezier-Bernstein polynomial, one can
successfully use them in both computer graphics and image data approxima-
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