The Saddle

Here is one view. The mesh curves are the slices with the vertical planes
x=-10,-9,...,9,10,    y=-10,-9,...,9,10.

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.