Dini's surface

Dini's surface with 0 ≤ u ≤ 4π and 0.01 ≤ v ≤ 1 and constants a = 1.0 and b = 0.2.

In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere.^[1] It is named after Ulisse Dini^[2] and described by the following parametric equations:^[3]

$x=a\cos\left(u\right)\sin\left(v\right)$

$y=a\sin\left(u\right)\sin\left(v\right)$

$z=a\left(\cos\left(v\right)+\ln\left(\tan\left(\frac{v}{2}\right)\right)\right)+bu$

Another description is a helicoid constructed from the tractrix.^[4]

Uses

Renditions of Dini's surface have appeared on the covers of Western Kentucky University's Graduate Study in Mathematics, Gray's Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd Edition), and in La Gaceta de la Real Sociedad Matemática Española (The Gazette of the Royal Spanish Mathematical Society), Vol. 2 (3).

See also

• Dini derivative
• Dini test
• Dini's theorem
• List of surfaces

References

1. "Wolfram Mathworld: Dini's Surface". Retrieved 2009-11-12.
2. J J O'Connor and E F Robertson (2000). "Ulisse Dini Biography". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 2016-04-12.
3. "Knol: Dini's Surface (geometry)". Retrieved 2009-11-12.
4. Rogers and Schief (2002). Bäcklund and Darboux transformations: geometry and modern applications in Soliton Theory. Cambridge University Press. pp. 35–36.