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researchers for its use in diverse areas. The success behind the ecient ap-
plications of this polynomial in many fields has also made it widely popular.
The basic philosophy behind the Bernstein polynomial approximation is that
this polynomial is very convenient to free-form drawing. In fact, some of the
properties of this polynomial are so attractive that no sooner than the tech-
nique was published by Bezier, it became widely popular in many industries.
In order to design the body of an automobile, Bezier developed a spline model
that became the first widely accepted spline model in computer graphics and
computer-aided design, due to its flexibility and ease over the then-used draw-
ing and design techniques. Since Bezier used the Bernstein polynomial basis
as the basis function in his spline model, the justification of the name “Bezier-
Bernstein” spline immediately applies and hence, the Bernstein basis domi-
nates the performance of the Bezier spline. This model, therefore, helps to
design and draw smooth curves and surfaces of different shapes and sizes,
corresponding to different arbitrary objects, based on a set of control points.
Bezier spline model, though is extensively used for free-form drawing, can
also be used to approximate data points originated from different functions.
The problem of function approximation is essentially the problem of estima-
tion of control points from a data set. Drawing and function approximation
are essentially different in nature, though approximation is done in both cases.
In the curve and surface design, approximation error is not of prime concern.
Visual effect or the aesthetics of the shape of the object is the sole objective.
So, one should observe how accurately a drawn object depicts the shape of
its corresponding target object. Notice that Bezier spline-based drawing tech-
nique starts from the zeroth order Bernstein approximation (which is exactly
the line drawing between control points) of the data points and goes to some
higher order (quadratic or cubic) approximation, until it mimics the shape
of the object. Step by step through interactions, a designer can make nec-
essary corrections to achieve perfection in shape of the object. On the other
hand, in a data approximation problem, we justify the approximation by the
error in approximation. This is a purely mathematical problem where we are
in no way concerned with the graphics involved behind the approximation.
Furthermore, if the data set corresponds to a graylevel image, the error in
approximation becomes subjective. We accept small or large error depending
on the nature of applications. Such an approximation of image data points is
useful in compression and feature extraction.
The concept of control points in Bezier-Bernstein spline is implicit in the
definition of the Bernstein polynomial and it was Bezier who made it explicit.
Later on, the concept of control points was generalized to knots in B-spline
to keep the interaction locally confined, so that the global shape of curves
and surfaces is least affected. The generalization, therefore, introduces more
drawing flexibility in the B-spline model.
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