When polar coordinate $\theta $ equals "co-latitude" $\gamma, $ the ruled surface generated has a constant negative Gauss curvature $ K=-\frac{1}{a^2}$ but only at the cuspidal edge $r=a.$ Only a half is shown for image clarity.
The parametrization is $$ (x,y,z)=( r \cos \theta, r \sin \theta, (a-r) \cot \theta ) $$
Is it possible to relate $\theta, \gamma $ so that $K$ is constant negative everywhere?
As this may not be possible, is there a curved generator connecting z-axis points to any other circle points in order to get the same result?
EDIT 1:
In a particular case (conic/hypo pseudosphere) K=-1 can be obtained using curved asymptotes on a surface of revolution connecting points on $r=a$ to a point on z-axis.
Can all such paths.. whether ruled or not ruled, whether on a surface of revolution or not, be found by a defining ode?
Thanks so much for all suggestions!
