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 [1]

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 [2] [3] [4] [5] [6] [7] [8].

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

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

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

[3] Andrew J.P. Maclean, “Parametric Equations for Surfaces”: http://public.kitware.com/VTK/pdf/VTKParametricSurfaces.pdf

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

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

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

[7] http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/dini/

[8] http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/dini/eqns.html

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About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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