Dini's surface

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Dini's surface plotted with adjustable parameters by Wolfram Mathematica program
Dini's Surface with constants a = 1, b = 0.5 and 0 ≤ u ≤ 4π and 0<v<1.

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 ⁡ u sin ⁡ v y = a sin ⁡ u sin ⁡ v z = a ( cos ⁡ v + ln ⁡ tan ⁡ v 2 ) + b u {\displaystyle {\begin{aligned}x&=a\cos u\sin v\\y&=a\sin u\sin v\\z&=a\left(\cos v+\ln \tan {\frac {v}{2}}\right)+bu\end{aligned}}} {\displaystyle {\begin{aligned}x&=a\cos u\sin v\\y&=a\sin u\sin v\\z&=a\left(\cos v+\ln \tan {\frac {v}{2}}\right)+bu\end{aligned}}}
Dini's surface with 0 ≤ u ≤ 4π and 0.01 ≤ v ≤ 1 and constants a = 1.0 and b = 0.2.

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

See also

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. Archived from the original on 2012-06-09. Retrieved 2016-04-12.
  3. "Knol: Dini's Surface (geometry)". Archived from the original on 2011-07-23. 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.