Tractricoid as a pseudosphere (surface with constant negative curvature) How to motivate/calculate/prove see that the tractricoid, i.e. a tractrix rotated about its asymptote, has a constant negative curvature?
What are the hyperbolic lines on a tractricoid and how to see that there are infinitely many parallel lines?
 A: Sorry not enough knowledge here but lets do an thought experiment:
As you now the tracioid is the rotation of a tractrix around its asymptope.
So lets first do the plane bit:
If you look at https://en.wikipedia.org/wiki/Tractrix 
you can see a picture of its evolute. 
the curvature of a curve is the reciprocal of the radius of the osculating circle 
https://en.wikipedia.org/wiki/Osculating_circle
for the tractrix the centre is is the point where the normal of the tractrix meets its evolute.
Now to the tracioid:
the curvature of a surface is the product of the maximum and minimal curvature of the osculating circles at a point.
see https://en.wikipedia.org/wiki/Principal_curvature and   https://en.wikipedia.org/wiki/Gaussian_curvature
for a point on the tracioid I think these ocilating circles are:


*

*the circle around its asymtope

*the circle trough its evolute 
and then just multiply them 
I was not able to find the right formulas that i should use here so this is more an thought experiment
I hope somebody else can elaborate on this (or show I am wrong , and how i should have done it) 
A: Tangent length $a$ upto asymptote x-axis is a constant =$a$.
This property of Tractrix can be  used to derive constant Gauss curvature: 
$$ \dfrac{ \sin \phi  }{r}= 1/a= \dfrac{ \cos \phi* d \phi /dl }{\sin\phi} $$
$$ \dfrac{ \sin ^2 \phi  }{r^2}= 1/a^2= \dfrac{ \cos \phi }{r}\cdot  d \phi /dl;\,  \kappa_2*\kappa_1 = -K= 1/a^2 $$
The second equality is obtained using Quotient Rule of differentiation with respect to meridian arc length l by cross multiplying last two  and dividing by $r^2.$
All asymptotic lines on the surface
$$ r/a = \sech(\theta),z/a = \theta - \tanh(\theta) $$
obtained by rotation around axis of symmetry $r=0$ and their reflections in $(r-z)$ plane constitute a set parallel lines,  parallel to each other,and also "parallel" to the axis of symmetry.
