Dini surface is a family of surfaces that have constant Gaussian curvature of -1. We write
K = -1.
(K = -1 surfaces are also called pseudospherical surfaces in general, because the unit sphere has constant Gaussian curvature of 1.)
Many explicitly parametrized surfaces of
K = -1
were found in the 1800s, among them the Dini family.
Today we have a general theory which associates with every soliton
solution of the sine-Gordon equation
q_xt = sin(q(x,t))
K = -1
surface. In this context the Dini family comes from the
q(x,t) = 4arctan(exp(ax + t/a).
are also called Kink solutions.
In hyperbolic space H^2 there are limits of circles with the radius going to infinity and one diameter endpoint is kept fixed. These limit circles are called horocircles and their insides are called horodisks.
The soliton solution
q(x,t) determines first and second fundamental form of the
(K = -1) surface. These give ODEs for the curvature lines.
Dini surface is also known as the one-soliton surface.
Dini Surface Parametric Equations x = (u - t) y = (r * cos(v)) z = (r * sin(v)) Where: psi := aa; if (psi < 0.001) then psi := 0.001; if (psi > 0.999) then psi := 0.999; psi := psi * pi; sinpsi := sin(psi); cospsi := cos(psi); g := (u - cospsi * v) / sinpsi; s := exp(g); r := (2 * sinpsi) / (s + 1 / s); t := r * (s - 1 / s) * 0.5;