Next, flex a piece of paper so that along the x-axis it matches r(x). Now any line parallel to the x-axis should match r(x). Let us tilt this on the y-axis, so our altitude function is h(x,y)= r(x) + y. Notice that we have a function whose second y-derivative vanishes everywhere. So is a PEF for it. Its Gaussian curvature hxxhyy-hxy2 vanishes. Now, to confuse people, we rotate it .Thus, I propose the test function h(x,y)= r(x+y) + (x-y). Its perfect PEF is
. 0 0 . 0 0 1 0 0 0 -2 0 0 0 1and we need to show that we can find it.
What will we give for data? I offer a track of dense data along the x axis (like a well log or seismogram). Already you can see with this and with the PEF, and one more data point off the track, the data everywhere is known. Well, almost everywhere. Actually, there is a shadow. Now we sprinkle a dozen data values around the plane. We might choose those points aliased on our rough function in such a way that everyone but us would be completely confused.