"Asymptotic " transformation for principal curvature lines on positive Gauss curvature K surfaces If we have lines of curvature for a  $ \mathbb R^2 $ surface in 3-space $ $(x,y,z)$  as
$$ [ f(u,v), \ g(u,v), \ h(u,v) ] $$ 
for negative Gauss curvature surfaces then  
$$  f(u+v,u-v), g(u+v,u-v) , h(u+v,u-v) $$
represent asymptotic lines whose normal curvature vanishes.
What is characteristic about the corresponding lines for positive K surfaces? What are its properties? I appreciate any remarks.. intuitive, qualitative, or equation-based regarding what curvature vanishes or remains constant, or basically why is it not considered important or interesting.
EDIT1:
How do lines rotate in tangent plane by virtue of such transformation in either case? Secondly are they conceptually linked to (eg Struik's Differential Geometry book) "Imaginaries in surface theory"?
 A: If I understand, you're asking:
Let $M$ be a surface in Euclidean $3$-space and $p$ a point of $M$. If the Gaussian curvature at $p$ is negative, then the asymptotic directions at $p$ can be expressed as "linear combinations" of the principal directions. What do the corresponding linear combinations means if the Gaussian curvature is positive at $p$?

Assuming that's correct:
Let $k_{1}$ and $k_{2}$ be the (non-zero) principal curvatures of $M$ at $p$. By a Euclidean motion, we may assume $p$ is the origin, and $T_{p}M$ is the $(x, y)$-plane with the coordinate axes as principal directions. In a sufficiently small neighborhood of $p$, the surface $M$ is a graph
$$
z = \tfrac{1}{2}(k_{1} x^{2} + k_{2}y^{2}) + o(x^{2} + y^{2}).
$$
For purposes of curvatures, we may neglect the higher-order terms and look at the quadratic approximation.
The asymptotic directions at $p$ are the lines of intersection (if any) of $\{z = 0\}$, the tangent plane at $p$, with the graph $z = \tfrac{1}{2}(k_{1} x^{2} + k_{2}y^{2})$. If the Gaussian curvature $k_{1} k_{2}$ is negative, put $k^{2} = -k_{1}/k_{2}$ and write
$$
0 = \tfrac{1}{2}(k_{1} x^{2} + k_{2}y^{2})
  = -\tfrac{1}{2} k_{2} (k^{2}x^{2} - y^{2})
  = -\tfrac{1}{2} k_{2} (kx - y)(kx + y).
$$
That is, the asymptotic directions are the lines $y = \pm kx$.
If the Gaussian curvature is positive, by contrast, the preceding factorization is non-real: Putting $k^{2} = k_{1}/k_{2}$, we have
$$
0 = \tfrac{1}{2}(k_{1} x^{2} + k_{2}y^{2})
  = \tfrac{1}{2} k_{2} (k^{2}x^{2} + y^{2})
  = \tfrac{1}{2} k_{2} (kx - iy)(kx + iy).
$$
Geometrically, the graph of a positive- or negative-definite quadric touches the tangent plane only at the point of tangency, so long as we work over the reals. In that sense, the corresponding directions are non-real, and therefore "uninteresting".
(Incidentally, if $M$ is real-analytic, then $M$ can be viewed locally as the "real points" of a complex surface in $\mathbf{C}^{3}$ by "allowing the power series variables to be complex". In this setting, the preceding factorization has geometric meaning. In retrospect there's no mystery, however, since all quadratic forms are equivalent over $\mathbf{C}$.)
