4.3.3 Matrix forms and change of basis Monomial or Power or Ferguson form R() =U·F M 33-2-1R() 0010 R(0) 1000 R(1) Conversion between the two representations is simply a matter of matrix manipulation U·F U·MH·FH F MH·F FH Ma·F 4.3, 4 Bezier curves Bezier developed a reformulation of Ferguson curves in terms of Bernstein polynomials for the UNISURF System at Renault in France in 1970. This formulation is expressed mathematically as follows R(u)=∑RBn()0≤u≤1 Ri and Bi, n(u) represent polygon vertices and the Bernstein polynomial basis functions respec tively. The definition of a Bernstein polynomial is Bi n(u) u2(1-u)-i=0,1,2, il(n-i) The polygon joining Ro, R1,..., Rn is called the control polygon Examples of Bezier curve n=2: Quadratic Bezier Curves(Parabola), see Figures 4.6 and Figure 4.7 R(u)=Ro(1-)2+R12u(1-)+R2 220R 00R24.3.3 Matrix forms and change of basis • Monomial or Power or Ferguson form R(u) = h u 3 u 2 u 1 i      a3 a2 a1 a0      = U · FM • Hermite-Coons R(u) = h u 3 u 2 u 1 i      2 −2 1 1 −3 3 −2 −1 0 0 1 0 1 0 0 0           R(0) R(1) R˙ (0) R˙ (1)      = U · MH · FH Conversion between the two representations is simply a matter of matrix manipulation: U · FM = U · MH · FH Hence, FM = MH · FH FH = M−1 H · FM 4.3.4 B´ezier curves B´ezier developed a reformulation of Ferguson curves in terms of Bernstein polynomials for the UNISURF System at Renault in France in 1970. This formulation is expressed mathematically as follows: R(u) = Xn i=0 RiBi,n(u) 0 ≤ u ≤ 1 Ri and Bi,n(u) represent polygon vertices and the Bernstein polynomial basis functions respec￾tively. The definition of a Bernstein polynomial is: Bi,n(u) = n i ! u i (1 − u) n−i i = 0, 1, 2, ...n where n i ! = n! i!(n − i)! The polygon joining R0, R1, ..., Rn is called the control polygon. Examples of B´ezier curves: • n=2: Quadratic B´ezier Curves (Parabola), see Figures 4.6 and Figure 4.7 R(u) = R0 (1 − u) 2 + R1 2u(1 − u) + R2 u 2 = h u 2 u 1 i    1 −2 1 −2 2 0 1 0 0       R0 R1 R2    7