Tractricoid as a pseudosphere (surface with constant negative curvature)

Question

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?

Answer 1

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.

Answer 2

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)