TITLE: Surface in 3D that realizes all pairs of principal curvatures
QUESTION [11 upvotes]: This is a question that Willie Wong raised in comments after he answered my question,
Surface analog of clothoid: curvatures covering $\mathbb{R}^3$. Willie's question is
more interesting (and challenging) than mine, essentially requiring the 
curvatures to map onto $\mathbb{R}^2$ rather than only onto $\mathbb{R}$:

Is there a surface $S \subset \mathbb{R}^3$ that realizes all pairs of principal curvatures in the sense that, for every $(\kappa_1,\kappa_2) \in \mathbb{R}^2$, 
  there is a point
  $p \in S$ such that the principal curvatures at $p$ are $\kappa_1$ and $\kappa_2$?

He also suggested that it might be natural to restrict to $\kappa_1 \ge \kappa_2$.

REPLY [7 votes]: Here is one view of Manfred's Angel's Curl surface, using $g(u) = e^{u^2}$:
          

Hopefully I've computed this correctly.
I've let the parameters $u$ and $v$ range over $\pm 4$.