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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.
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