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.

Converted by