Here is one view. The mesh curves are the slices with the vertical planes
Here is a side view of the same saddle:
Every point p=(a,b,c) on the saddle is contained in exactly two lines that lie on the saddle.
Given p=(a,b,c), these lines have parametric equations
p+t (1,1,2a-2b) and p+t(1,-1,2a+2b), respectively.
The lines are the wounds in the saddle caused by slicing with the plane z=2ax-2by-c.
In the first picture, we take p=(0,0,0), the slicing plane is the xy plane (z=0),
and the lines are the lines x=y, x=-y in the xy-plane. The mesh curves are hidden.
In the second picture, we take p=(5,1,24). The slicing plane is z=10x-2y-24.
In the last picture, we show the same saddle, drawn in Polar coordinates.
The mesh curves are the slices with the planes theta=constant.
You can see that two of these mesh curves are the lines on the saddle through
(0,0,0), as seen above.