They form a basis for polynomials (see Sect.
4.4) and have several properties of
interest:
Non-negativity:
.
Partition of unity:
(by the binomial theorem).
Symmetry:
(1.20)
Recursion:
with
for
0,
and
.
Linear precision:
(1.21)
which implies that the monomial
can be expressed as the weighted
sum of Bernstein polynomials of degree
with coefficients evenly
spaced in the interval [0,1]. This property is used extensively in
Chaps. 4 and 5.
Degree elevation: The basis functions of degree
can be expressed in terms of those of degree
[106] as:
(1.22)
where
.
Or more generally in terms of basis functions of degree
[106] as:
(1.23)
Figure 1.2 shows the Bernstein polynomials of degree 3
and 4. The derivative of a Bernstein
polynomial is
(1.24)
where
.
Figure 1.2:
Bernstein polynomials: (a) degree three, (b) degree four
Next: 1.3.2 Arithmetic operations of
Up: 1.3 Bézier curves and
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December 2009