eval_cshep2d - bidimensional cubic shepard interpolation evaluation
This is the evaluation routine for cubic Shepard interpolation function computed with cshep2d , that is :
zp(i) = S(xp(i),yp(i)) dzpdx(i) = dS/dx(xp(i),yp(i)) dzpdy(i) = dS/dy(xp(i),yp(i)) d2zpdxx(i) = d2S/dx2(xp(i),yp(i)) d2zpdxy(i) = d2S/dxdy(xp(i),yp(i)) d2zpdyy(i) = d2S/dy2(xp(i),yp(i))
The interpolant S is C2 (twice continuously differentiable) but is also extended by zero for (x,y) far enough the interpolation points. This leads to a discontinuity in a region far outside the interpolation points, and so, is not cumbersome in practice (in a general manner, evaluation outside interpolation points (i.e. extrapolation) leads to very inacurate results).
// see example section of cshep2d // this example shows the behavior far from the interpolation points ... deff("z=f(x,y)","z = 1+ 50*(x.*(1-x).*y.*(1-y)).^2") x = linspace(0,1,10); [X,Y] = ndgrid(x,x); X = X(:); Y = Y(:); Z = f(X,Y); S = cshep2d([X Y Z]); // evaluation inside and outside the square [0,1]x[0,1] m = 40; xx = linspace(-1.5,0.5,m); [xp,yp] = ndgrid(xx,xx); zp = eval_cshep2d(xp,yp,S); // compute facet (to draw one color for extrapolation region // and another one for the interpolation region) [xf,yf,zf] = genfac3d(xx,xx,zp); color = 2*ones(1,size(zf,2)); // indices corresponding to facet in the interpolation region ind=find( mean(xf,"r")>0 & mean(xf,"r")<1 & mean(yf,"r")>0 & mean(yf,"r")<1 ); color(ind)=3; xbasc(); plot3d(xf,yf,list(zf,color), flag=[2 6 4]) legends(["extrapolation region","interpolation region"],[2 3],1) xselect()
cshep2d ,