# Conic K=-1 Surface of Revolution

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

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 skew-symmetric solutions defined by initial conditions

• `q(0) = 0`
• `q'(0) = b > 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. An anaglyph wire frame sequence of the same deformation as before. The final surface has screw motion symmetry and no self intersections. conic k 1 sor st conic k 1 sor sw

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Conic_K=_-1_Surface_of_Revolution.pdf