[The Beltrami pseudosphere][1]

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

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 ?


  [1]: http://virtualmathmuseum.org/Surface/pseudosphere/pseudosphere.html