The Bianchi-Bäcklund transform leads to more and more complicated, but
explicit, solutions q(u,v) of the Sine-Gordon Equation (SGE). Each SGE
solution determines a parametrized surface `F(u,v)`

with Gauss curvature
`K = -1`

.

In spite of the sharp rims these surfaces are defined for all `(u,v)`

in R^2. The singular rims occur in a similar way as the cusps of rolling
curves: At the zeros of `sin(q(u,v))`

the velocity of one set
of parameter lines becomes zero and the parameter lines reverse their direction,
so that each produces a cusp and all these cusps are a rim.