# Breather Surface

The Breather surface is the surface of Gaussian
curvature -1 that corresponds to the
following solution of the Sine-Gordon equation:

`q(u,v) := 4 * arctan( (aa / w) * (sin(w * v)) / (cosh(aa * u)) )`

where:

`w := sqrt( 1 - aa * aa)`

Breather Surface can be parametrized by:

wsqr := 1 - aa * aa;
w := sqrt(wsqr);
denom := aa * (sqr(w * cosh(aa * u)) + sqr(aa * sin(w * v)))
x = -u + (2 * wsqr * cosh(aa * u) * sinh(aa * u) / denom)
y = 2 * w * cosh(aa * u) * (-(w * cos(v) * cos(w * v)) - (sin(v) * sin(w * v))) / denom
z = 2 * w * cosh(aa * u) * (-(w * sin(v) * cos(w * v)) + (cos(v) * sin(w * v))) / denom

(Note: curvature lines, are curves on the surface that are formed by the 2 directions of principle curvatures, at each point.)

## Anaglyph

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DiniKuenBreather.pdf