I have a parameterized surface $h$, defined on an orthogonal (u,v) plane, described by: $$ h(u,v)=\sqrt{(au+bv+c)^2+(du+ev+f)^2} $$ where $a...f$ are parameters. I’m choosing $a...f$ so that the surface is well behaved: smooth, no negative roots or discontinuities. The surface forms a simple valley, and the valley floor seems to be a sloping straight line when I plot the function.
My question is: what is the relationship between $u$ and $v$ describing the valley in the $(u,v)$ plane, i.e. how do I find the valley floor (principal curvature?) and then project it into the $(u,v)$ plane (red line in the plot)?
In other words, if there is just a simple linear relationship between $u$ and $v$ describing the valley floor, i.e. $u(v)=mv+n$, how can I find $m$ and $n$ from the parameters $a…f$? plot of $h(u,v)$
