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I have a problem while finding the equation of a tangent plane to a parametric surface. The surface is given by $$\pmb r = \begin{bmatrix} s^2 + t^2 \cr 2s+2t \cr 2 \end{bmatrix}$$

The point at which plane is tangent $$ \pmb r_o = \begin{bmatrix} 2 \cr 4 \cr 2 \end{bmatrix} (s=1, t=1)$$

I know that for tangent-plane equation, I need to find a normal vector $\pmb n$ by doing a cross-product of $\frac{\partial \pmb r}{\partial t}$ at $r_o$ and $\frac{\partial \pmb r}{\partial s}$ at $r_o$.

My problem is that both $\frac{\partial \pmb r}{\partial t}$ and $\frac{\partial \pmb r}{\partial s}$ are being computed as the same vector $\begin{bmatrix} 2 \cr 2 \cr 0 \end{bmatrix}$. So my normal vector will always be zero, and I can't use it to find the equation of the tangent.

How to get around this problem?

Also, in general, how do we solve tangent-plane equations to parametric surfaces in such cases?

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Your "surface" $S$ is the subset $x\geq y^2/8$ inside the plane $z=2$. As such, it is not regular along the curve $(\tau, \tau^2/8, 2)$ and, according to most standard definitions, one cannot talk about tangent planes at points along that curve. Since $\pmb r(1,2)$ lies on this curve, we have no well-defined tangent. Of course, since $S$ is entirely contained in the plane $z=2$, at any point not on its boundary curve the tangent plane is $z=2$.

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  • $\begingroup$ Okay. Thanks! Is there a way to identify curves along which a given surface is regular? Also, can a surface be non-regular too if the surface function is a scalar-valued function? $\endgroup$ Dec 20 '20 at 12:30
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    $\begingroup$ In general, no easy way that I know. You can identify the points along which the given parameterization is not regular, but then for each of them you have to decide if it is just a bad parameterization for/near that point, or if the surface itself is not regular at that point. This can be more challenging (but can sometimes be done - once you understand the surface well enough - using the implicit function theorem). $\endgroup$
    – Max
    Dec 20 '20 at 12:35

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