Graphics Reference
In-Depth Information
1
Bernstein Polynomial and Bezier-Bernstein
Spline
1.1 Introduction
Bernstein polynomial, its significance, different properties, and detection of its
order for approximation of a data set, are very important and useful as a first
course material to study splines. In fact, Bernstein polynomial can be thought
of as the gateway to splines, namely the Bezier spline. Its strong relation with
the Bezier spline can, in no way, be forgotten. Bezier polynomial can be made
to act in either of these ways: as a spline or as a non-spline. When it acts as
a spline, it does piecewise approximation of a data set with some smoothness
conditions satisfying at the break points, but when it acts as a non-spline to
approximate, it does not take into consideration the smoothness conditions
to satisfy at the break points. Readers interested in details of Bernstein poly-
nomial may consult any standard text book on mathematics. Bezier curves,
on the other hand, show how their geometry is influenced by Bernstein poly-
nomials. As Bezier curves and surfaces are driven by Bernstein basis, they
can also be thought of, respectively, the Bernstein polynomial pieces of curves
and surfaces. P. E. Bezier, a French designer in the automobile industry for
Renault, suggested a revolutionary concept for the interactive design of curves
and surfaces. He suggested that these curves behave exactly the same way as
humans do until satisfaction reaches a maximum. For this, he artfully incor-
porated [22] the Bernstein basis and some control points in his design. This
concept of control points and their positioning play the most significant and
vital role in his interactive design mechanism.
1.2 Significance of Bernstein Polynomial in Splines
Bernstein polynomial is well known in the mathematical theory of function
approximation. It can be used to approximate known, as well as unknown,
functions with any desired degree of accuracy. Besides, this polynomial pos-
sesses a number of significant properties that have made it attractive to many
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