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).

enter image description here

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.

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    $\begingroup$ Tempted to give this question kudos for one of the coolest question headlines I've seen in a while. I asked my wife that question, and she went into great detail about subject matter I choose not to discuss on the web. $\endgroup$ Mar 20, 2015 at 22:58
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    $\begingroup$ The paraboloid should (explicitly) be taken relative to the point you're interested in - you're tracking the shape of your surface around that point of contact. Your osculating paraboloid should probably be $g(x,y) = \frac12(L_{11}(x-p_x)^2+2L_{12}(x-p_x)(y-p_y)+L_{22}(y-p_y)^2$. Think of it as being analagous to taking the Taylor series of a univariate function around a point (since that is, after all, the direct analog!). $\endgroup$ Mar 20, 2015 at 23:48
  • $\begingroup$ @Steven -- Thanks, I feel like it must be something similar to this. I've tried your suggestion a few different ways ($x+p_x$, $x-p_x$ and $p_x-x$) and get similarly close-but-no-cigar results... $\endgroup$
    – helveticat
    Mar 21, 2015 at 8:15
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    $\begingroup$ @helviticat The issue is that the paraboloid needs to be expressed as a graph of a function relative to the tangent plane (i.e., you have to pick coordinates (u,v) that make the tangent plane at the point in question "horizontal" and the normal vector at said point vertical). There are a number of ways to do this (and I don't know which is most efficient offhand). This is a similar issue to if you tried to plot the oscillating circle to the graph of a curve in three-dimensional space; it isn't enough to put the approximation at the point in question, it also needs to be oriented correctly. $\endgroup$
    – THW
    Mar 22, 2015 at 20:37
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    $\begingroup$ Let me know if you would like more details as an answer. If I have the time, I will happily do so. $\endgroup$
    – THW
    Mar 22, 2015 at 20:44


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