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202 C. Liang et al.
[a 2 ,b 2 ] ⊂ R , its x-face (resp. y-face, z-face) are its faces normal to the direction x
3
(resp. y, z).
11.2.1 Univariate Bernstein basis
Given an arbitrary univariate polynomial function f(x) ∈ K, we can convert it into
the representation of Bernstein basis of degree d, which is defined by:
d
f(x)= b i B (x), and (11.1)
i
i
d i d−i
d
B (x)= x (1 − x) (11.2)
i i
where b i is usually referred as controlling coefficients. Such conversion is done
through a basis conversion [6]. The above formula can be generalized to an arbi-
trary interval [a, b] by a variable substitution x =(b − a)x + a. We denote by
d
i
i
B (x; a, b) (x − a) (b − x) d−i (b − a) −d the corresponding Bernstein basis on
d
i
[a, b].
There are several useful properties regarding Bernstein basis given as follows:
i i
• Convex-Hull Properties: Since B (x; a, b) ≡ 1 and ∀x ∈ [a, b],B (x; a, b) ≥
i d d
0 where i =0, ..., d, the graph of f(x)=0, which is given by (x, f(x)), should
always lie within the convex-hull defined by the control coefficients [5].
• Subdivision (de Casteljau): Given t 0 ∈ [0, 1], f(x) can be represented piece-
wisely by:
d d
i
i
f(x)= b B (x; a, c)= b (d−i) B (x; c, b), where (11.3)
(i)
i
d
d
0
i=0 i=0
b (k) =(1 − t 0 )b (k−1) + t 0 b (k−1) and c =(1 − t 0 )a + t 0 b. (11.4)
i i i+1
Another interesting property of this representation is related to Descartes’ Law
of signs. The definition of Descartes’ Law for a sequence of coefficients
b k = b i |i =1, ..., k
is defined recursively:
1, if b i b i+1 < 0
V (b k+1 )= V (b k )+ (11.5)
0, else
With this definition, we have:
n d
Theorem 1. Given a polynomial f(x)= b i B (x; a, b), the number N of real
i i
roots of f on ]a, b[ is less than or equal to V (b), where b =(b i ),i =1, ..., n and
N ≡ V (b)mod 2.