$$[x = a \sin p \cos t , y= - a ( \cos p + \log \tan p/2 ) , z= b+ a \sin p \sin t \; ], (.1 <p<\pi/2), (0< t< 2 \pi), \; (b>a) $$
is bent to a non-axisymmetric surface so that its straight axis of symmetry along $z=b$ before bending goes to circle $ y^2+z^2 = b^2 $ to radius $b$ in an isometric mapping preserving its Gauss curvature $ K=-1/a^2 $. The plot is drawn for $a=1,b=2$ units.
EDIT1:
Bending is such that 1) Opposites points of a diameter on the meridian in $yz$ plane before deformation remain in the $yz$ plane and 2) Normals intersecting on pseudosphere straight axis before deformation intersect on circle $ y^2+z^2= b^2 $ after deformation as sketched.
How is a parametrization of bent surface found ?
It is expected that the cuspidal circle to distort and smaller tubular part getting split as positive $K$ cannot develop in a closed toroidal configuration.
