I need the unit normal vector to this surface for various computations but I am finding it very hard to work with it:
$$\mathbf{x}(u,v) = r(u) + a(v)\mathbf{n} + b(v)\hat{z}$$
where $r,s$ are unit speed curves on the plane $z=0$, $\hat{z}$ is the unit vector in the z-direction and $s(t) = a(t)\hat{x} + b(t)\hat{y}$. Also $\mathbf{n}$ is the normal to the curve $r$.
What I have is
$$ \mathbf{x_u} = r' + a\mathbf{n}' \quad \mathbf{x_v} = a'\mathbf{n}+b'\hat{z} $$
$$N = \frac{\mathbf{x_u}\times \mathbf{x_v}}{|\mathbf{x_u}\times \mathbf{x_v}|}$$
But
$$\mathbf{x_u}\times \mathbf{x_v} = (r' \times a'\mathbf{n})+ (r' \times b'\hat{z})+ (a\mathbf{n}'\times a'\mathbf{n})+ (a\mathbf{n}'\times b'\hat{z}) $$
I don't know how to continue from here. How can I simplify this to work with it and get something for $N$
I recognize that $(r' \times a'\mathbf{n})$ is normal to the plane $z=0$ whereas the rest of the terms are on the plane. Not sure how that helps.
In the end my goal is to compute the Gaussian curvature. For that I need the second fundamental form and for that I need some reasonable $N$ to work with.
Otherwise I'm not sure how to compute $\langle N,\mathbf{x_{uu}}\rangle$.
Alternatively is it possible to take the coordinates $\hat{r}, \hat{n},\hat{z}$ so that $\mathbf{x} = (r(u), a(v), b(v))$?
\begin{align*} \mathbf{t}(u) &= \mathbf{r}' \\ \mathbf{t}'(u) &= \kappa \mathbf{n} \\ \mathbf{n}'(u) &= -\kappa \mathbf{t} \\ \mathbf{t}(u) \times \mathbf{n}(u) &= \mathbf{e}_z \\ |\dot{\mathbf{s}}(v)| &= 1 \\ \sqrt{\dot{a}(v)^2+\dot{b}(v)^2} &= 1 \\ \mathbf{x}(u,v) &= \mathbf{r}(u)+a(v) \mathbf{n}(u)+b(v) \mathbf{e}_z \\ \partial_u \mathbf{x} &= \mathbf{r}'(u)+a(v) \mathbf{n}'(u) \\ &= [1-\kappa a(v)] \mathbf{t} \\ \partial_v \mathbf{x} &= \dot{a}(v) \mathbf{n}+\dot{b}(v) \mathbf{e}_z \\ \partial_u \mathbf{x} \times \partial_v \mathbf{x} &= [1-\kappa a(v)][-\dot{b}(v) \mathbf{n}+\dot{a}(v) \mathbf{e}_z] \\ \mathbf{N} &= \frac{\partial_u \mathbf{x} \times \partial_v \mathbf{x}} {|\partial_u \mathbf{x} \times \partial_v \mathbf{x}|} \\ &= \frac{-\dot{b}(v) \mathbf{n}+\dot{a}(v) \mathbf{e}_z} {\sqrt{\dot{a}(v)^2+\dot{b}(v)^2}} \\ &= -\dot{b}(v) \mathbf{n}+\dot{a}(v) \mathbf{e}_z \end{align*}