From curves to cylinders

We know that from a planar curve $\alpha:I\rightarrow{\mathbb R}^2$ we can define the generalized cylinder as $S=\{(\alpha(s),t): s\in I, t\in{\mathbb R}\}$. This can also view as follows. 

Consider a smooth function $f:O\subset{\mathbb R}^2\rightarrow {\mathbb R}$ and $a\in {\mathbb R}$ a regular value. Then we know that $C=f^{-1}(\{a\})$ is a regular curve: the $1$-dimensional case than a surface. With this function $f$, we define a new function $F$ as $$F:O\times{\mathbb R}\rightarrow{\mathbb R},\ F(x,y,z)=f(x,y),$$

which is differentiable. We prove that $a$ is a regular value of $F$. Indeed, $$\nabla F=(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z})=(\nabla f,0).$$ Consequently, $(x,y,z)$ is a critical point of $F$ if and only if $(x,y)$ is a critical point of $f$. In particular, if $F(x,y,z)=a$, then $f(x,y)=a$ and $(x,y)$ is not a critical point of $f$, and thus, $(x,y,z)$ is not a critical point of $F$. This proves that $a$ is a regular value of $F$. 

As a consequence, $S=F^{-1}(\{a\})$ is a surface. We describe this surface: $$S=\{(x,y,z)\in O\times {\mathbb R}: f(x,y)=a\}=C\times {\mathbb R}.$$ This shows that $S$ is the generalized cylinder over the planar curve $C$.

The next picture is a generalized cylinder based on the cardioid.