# Parametrization of twisted pseudospheres types 2 and 3

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

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.

• You may be interested in ResourceFunction["GeneralizedHelicoid"] and ResourceFunction["TwistedSurface"] . I can't say if there's anything in there that solves your problem, but it may help simplify any code you write. There is also SurfaceData["DiniSurface", {"ParametricEquations", "GaussianCurvature"}] – flinty Oct 3 '20 at 13:09