MATLAB_Files.zip_nurbs

function [yy,dy,d2] = natspline(x,y,xx) % natspline: natural spline with differentiation % [yy,dy,d2] = natspline(x,y,xx): uses a natural cubic spline % interpolation to find yy, the values of the underlying function % y at the points in the vector xx. The vector x specifies the % points at which the data y is given. % input: % x = vector of independent variables % y = vector of dependent variables % xx = vector of desired values of dependent variables % output: % yy = interpolated values at xx % dy = first derivatives at xx % d2 = second derivatives at xx n = length(x); if length(y)~=n, error('x and y must be same length'); end if any(diff(x)<=0),error('x not strictly ascending'),end m = length(xx); b = zeros(n,n); aa(1,1) = 1; aa(n,n) = 1; %set up Eq. 18.27 bb(1)=0; bb(n)=0; for i = 2:n-1 aa(i,i-1) = h(x, i - 1); aa(i,i) = 2 * (h(x, i - 1) + h(x, i)); aa(i,i+1) = h(x, i); bb(i) = 3 * (fd(i + 1, i, x, y) - fd(i, i - 1, x, y)); end c=aa\bb'; %solve for c coefficients for i = 1:n - 1 %solve for a, b and d coefficients a(i) = y(i); b(i) = fd(i + 1, i, x, y) - h(x, i) / 3 * (2 * c(i) + c(i + 1)); d(i) = (c(i + 1) - c(i)) / 3 / h(x, i); end for i = 1:m %perform interpolations at desired values [yy(i),dy(i),d2(i)] = SplineInterp(x, n, a, b, c, d, xx(i)); end end function hh = h(x, i) hh = x(i + 1) - x(i); end function fdd = fd(i, j, x, y) fdd = (y(i) - y(j)) / (x(i) - x(j)); end function [yyy,dyy,d2y]=SplineInterp(x, n, a, b, c, d, xi) for ii = 1:n - 1 if xi >= x(ii) - 0.000001 & xi <= x(ii + 1) + 0.000001 yyy=a(ii)+b(ii)*(xi-x(ii))+c(ii)*(xi-x(ii))^2+d(ii)... *(xi-x(ii))^3; dyy=b(ii)+2*c(ii)*(xi-x(ii))+3*d(ii)*(xi-x(ii))^2; d2y=2*c(ii)+6*d(ii)*(xi-x(ii)); break end end end