05CurveFitting.ppt

* * * * * * * * * * * * * * * * * * * * * * * * 5.4 Fourier Series and Trigonometric Polynomials Functions g(x) are periodic Period 2π Cos(jx) and sin(jx). Can a periodic function be represented by sum of them.??? (P230)Def. 5.2 piecewise continuous Def. 5.3 Fourier series 5.4 Fourier Series and Trigonometric Polynomials Theorem. 5.5 Fourier Expansion 5.4 Fourier Series and Trigonometric Polynomials (P232)Theorem. 5.6 Cosine Expansion Theorem. 5.7 Sine Expansion 5.4 Fourier Series and Trigonometric Polynomials (P234)Def. 5.4 Trigonometric Polynomial Theorem. 5.8 Discrete Fourier Series 5.4 Fourier Series and Trigonometric Polynomials Trigonometric Polynomial Approximation function [A B]=tpcoeff(X,Y,M) %Input - X is a vector of equally spaced abscissas in [-pi pi] % - Y is a vector of ordinates % - M is the degree of the trigonometric polynomial %Output - A is a vector containing the coefficients of cos(jx) % - B is a vector containing the coefficients of sin(jx) N=length(X)-1; max1=fix((N-1)/2); if Mmax1 M=max1; end A=zeros(1,M+1); B=zeros(1,M+1); 5.4 Fourier Series and Trigonometric Polynomials code Yends=(Y(1)+Y(N+1))/2; Y(1)=Yends; Y(N+1)=Yends; A(1)=sum(Y); for j=1:M A(j+1)=cos(j*X)*Y; B(j+1)=sin(j*X)*Y; end A=2*A/N; B=2*B/N; A(1)=A(1)/2; 5.5 Bezier Curve (P239)Def. 5.5 Bernstein polynomials Degrees 1,2,3 Properties of Bernstein Polynomials (P239)Prop. 1 Recurrence Relation Prop. 2 Nonnegative on [0,1] Prop. 3 Form a partion of unity Prop. 4 Derivatives 5.5 Bezier Curve Properties of Bernstein Polynomials (P241)Prop. 5 Basis Def. 5.6 Parametrically as 5.5 Bezier Curve Properties of Bezier Polynomials Prop. 1 The points P0 and P1 are on the curve P(t) t=0 t=1 5.5 Bezier Curve Properties of Bezier Polynomials Prop. 2 P(t) is continuous and derivative on [0,1] 5.5 Bezier Curve Properties of Bezier Polynomials Prop. 3 Def. 5.7 The convex hull of a s