## Dini’s Surface (geometry)

A surface of constant negative curvature obtained by twisting a pseudosphere is known as the Dini’s Surface. It is named after Ulisse Dini.

The parametric equations are:

x = a cos(u) sin(v)
y = a sin(u) sin(v)
z = a{cos(v) + ln[tan(v/2)]} + bu

where

0 <= u <= 2*pi
0 < v < pi

If we take   a=1, b=0.2,

the figure looks like

For an interactive exploration of the Dini’s surface see the reference 

This version of Dini’s Surface appeared on the cover of the Graduate Study in Mathematics, Western Kentucky University

For in-depth coverage of the Dini’s Surface please refer to the references       .

Dini’s surface can also be found on the the excellent web site called Mathematical Imagery by Jos Leys.

 J.T.J. Dodson’s Java version of Dini’s Surface: http://www.maths.manchester.ac.uk/~kd/geomview/dini.html

 Dini’s Surface (Wolfram Mathworld) : http://mathworld.wolfram.com/DinisSurface.html

 Andrew J.P. Maclean, “Parametric Equations for Surfaces”:

 William P. Thurston, “The Geometry and Topology of Three-Manifolds”, Lecture notes from Princeton University: http://www.msri.org/publications/books/gt3m/

 Paul Bourke, Twisted Pseudosphere: http://local.wasp.uwa.edu.au/~pbourke/geometry/dini/

 J.T.J. Dodson on Geometry: http://www.maths.manchester.ac.uk/~kd/homepage/dodson.htm 