What Is the Parallel of Least Radius on a Hyperboloid of Revolution?

[Dragonfall]

Homework Statement

Show that on of the hyperboloid of revolution x^2+y^2-z^2=1, the parallel of least radius is the line of striction, ...


What's the parallel of least radius?

 

[HallsofIvy]

The "parallel of least radius" is the line, parallel to an axis, that is shortest from one point on the hyperboloid to another.

Now, clear up my confusion: what is a "line of striction"?

 

[Dragonfall]

Given a ruled surface x(t,v)=a(t)+vw(t), a line of striction is a curve b(t) such that <b'(t),w'(t)>=0 for all t and b lies on the trace of x, ie b(t)=a(t)+u(t)w(t) for some real valued function u(t). It be can then shown that u(t) is given by

$$u=-\frac{<a',w'>}{<w',w>}$$.

The points of a line of striction are the "central points" of the ruled surface.