$$[x = a \sin p \cos t , y= - a ( \cos p + \log \tan p/2 ) , z= b+ a \sin p \sin t \; ], (.1 <p<\pi/2), (0< t< 2 \pi), \; (b>a). $$
is bent 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 $.
How is a new parametrization found ?