Equations for Dini's Surface

The parametric equations for Dini's Surface are:

f(u,v) = (cos(u)sin(v),sin(u)sin(v),cos(v)+log(tan(v/2))+a*u),

0 <= u <= 2*pi, 0 < v < pi.

Notice that when v goes to zero and to pi, the z coordinate of Dini's surface goes to negative infinity and to positive infinity respectively. In the picture, c <= v <= pi/2 , where c is chosen arbitrarily. This parameter "c" cuts off the peak of Dini's surface.

With this parameterization we got:

Notice that the mesh is compressed in the top, therefore we did a reparameterization in order to get a more uniform mesh.

The new parameterization is:

d = atan(-m) + pi/2 

w = atan(2m*v/pi - m) + pi/2 

v1 = (pi-2c)(w-d)/(pi-2d) + c 

f(u,v1) = (cos(u)sin(v1),sin(u)sin(v1),cos(v1)+log(tan(v1/2))+a*u)

Where "m" controls how much the mesh is compressed or stretched.

After this parameterization we got:

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