Dini Surface

Dini Surface is a singly periodic surface of constant negative Gaussian curvature, given by a parametrisation, embedded, with a continuous symmetry group.

Open Dini Surface in the interactive viewer →

constant-curvature def-parametric embedded implemented k-const-negative singly-periodic tradition-classical

About

Take the pseudosphere and drag it along a helix as it turns: the result keeps the same constant negative curvature but winds around an axis, which is why it is often drawn as a twisted horn. Constant curvature survives the twisting because the motion is a screw, and a screw is a rigid motion.

Formula

x(u, v)

\sqrt{1 - 0.2^{2}} \cos\left(u\right) \sin\left(v\right)

y(u, v)

\sqrt{1 - 0.2^{2}} \sin\left(u\right) \sin\left(v\right)

z(u, v)

\sqrt{1 - 0.2^{2}} \left(\cos\left(v\right) + \log\left(\tan\left(\frac{v}{2}\right)\right)\right) + 0.2 u

Properties

Family: constant-curvature
Given by: a parametrisation
Curvature: constant negative Gaussian curvature
Periodicity: singly periodic
Ends: 0
Embedding: embedded
Symmetry kind: continuous symmetry group
Fidelity: exact
Exactness: elementary

Sources

Chart as curated in tools/surfdb/charts.py, verified numerically against the curvature condition the record claims (measured over the chart by the validator and the charts self-test).

U. Dini, 'Sopra le funzioni di una variabile complessa' and the helicoidal surfaces of constant curvature (1865).