Hyperbolic K=-1 Surface of Revolution

hyperb Keq 1 revol 001
Hyperbolic K = -1 Surface of Revolution

One half of these surfaces (cut in the middle) converge to the Pseudosphere

hyperbolic k1 dinied 001
The formulas for the meridians of the surfaces of revolution with Gauss curvature K = -1 are explicit and well known. To deform these surfaces, keeping K = -1, it is more convenient to construct these surfaces from solutions of the Sine-Gordon equation (SGE). For surfaces of revolution these SGE solutions can be obtained from the ODE q''(u) = sin(q(u)), with symmetric solutions defined by initial conditions q(0) = h > 0 and q'(0) = 0. Then one can do the same as in the classical Dini deformation: define new solutions qn(u,v) := q(cosh(d)*u + sinh(d)*v), with d the Dini parameter. The animation shows these surfaces. Most of them have self intersections, but some have screw motion symmetry - see the last image of the animation.
hyperbolic k1 sor dini
This is a longer piece of the last surface in the previous animation.
hyperbolic k1 sor st
hyperbolic k1 sor st
hyperbolic k1 sor sw
hyperbolic k1 sor sw