Hi Brian,
""""""""""Could you please explain what you mean by "end tangent to the plane" ?""""""""""

Where the ends conclude pointing tangent to the direction. Here is a picture with the start and end points tangent to the direction.

The end tangent to the plane "horizontally" occurs at local minimum values and local maximum values, per
formulas here:
http://calculus.subwiki.org/wiki/Sinc_function#First_derivative

(tan x = -x or tan x = x and x an element of intervals.....)

Need some more study...