Parametrization of twisted pseudospheres types 2 and 3

Question

Among the three types of rotationally symmetric Pseudospherical surfaces $K=-1$ (Beltrami central, type 2 and type 3)

(http://xahlee.info/surface/gallery.html)

we have Dini twist addition term $ b\cdot \theta$ on the first classical

$$(x,y,z)=( a \text{ sech} u \cos \theta,\; a(\theta- \tanh \theta)+ b\cdot \theta,\;a \text{ sech} u \sin \theta) ;$$

that changes $K$ from (when $b=0$) $-1/a^2$ to $\dfrac{-1}{a^2+b^2}$.

Attempting to find twisting parametrizations for types 2 and 3 by the same additional term to result in constant $K.$

From the definition of Gauss curvature we have the ODE primed on arc length $s$ as independent variable

$$ \;\kappa_1= \phi' ,\kappa_2-=\frac{\cos \phi}{r}, K= \kappa_1\kappa_2=\frac{\cos \phi\; \phi'}{r}=1/a^2$$

For visualisation purpose a numerical solution is attempted (instead of using closed form elliptic integrals representing the meridian) in order to parametrize twisted and type 2 (elliptic/conic) type 3 (hyperbolic) surfaces. Term $ b\cdot \theta$ added for Dini twist modification.

Gauss curvature is computed in the normal way, ( in terms of first and second fundamental form quotients ) and plotted below. However, it is found to vary.

It appears that this Dini twist term addition cannot work for types 2 and 3, but works only for the central Beltrami PS.

Please help to find the correct parametrization of the type 2 and 3 pseudospheres.

Thanking you in anticipation.