I'm trying to learn about the Second Fundamental Form and I thought it would be fun to set up a surface in Geogebra and try to calculate the osculating paraboloid as I moved a point around on it, for certain values of "fun". I'm using Banchoff & Lovett's textbook.
The red surface is parameterized by the horizontal grey xy-plane as $f(x, y) = (x, y, x^3-3xy^2)$. I've calculated the second fundamental form, I think correctly since I did it myself and my result matches what's in the book (it's a worked example).
The osculating paraboloid, I thought, should be $g(x, y) = 1/2(L_{11}x^2 + 2L_{12}xy + L_{22}y^2)$. This is plotted as the greenish-yellow surface.
As you can (perhaps) see, my "osculating paraboloid" doesn't osculate very nicely. In fact, the point p, at which it's calculated, isn't even on it at all. Nevertheless, when I move p around I can see the paraboloid doing something sensible, and "nearly right".
I've looked in multiple references and although I find the same formula repeated many times I haven't been able to figure out what crucial part I'm missing.
In case anyone's interested in playing with the Geogebra file, here's a link (Dropbox). To move p, click and drag q (in the coordinate plane); everything else is self-explanatory I think.