Catalan

Catalan is a minimal surface, given by a parametrisation, immersed.

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aperiodic def-parametric immersed implemented minimal tradition-classical

Formula

x(u, v)

u - \sin\left(u\right) \cosh\left(v\right)

y(u, v)

1 - \cos\left(u\right) \cosh\left(v\right)

z(u, v)

4 \sin\left(\frac{u}{2}\right) \sinh\left(\frac{v}{2}\right)

Properties

Family: minimal
Given by: a parametrisation
Curvature: zero mean curvature
Periodicity: not periodic
Ends: 0
Embedding: immersed
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).

E. Catalan (1855), H. F. Scherk (1835), H. W. Richmond (1904).
The catalog follows Matthias Weber's repository of minimal surfaces, https://minimalsurfaces.blog/ , and his lecture notes "Classical Minimal Surfaces in Euclidean Space by Examples" (Indiana University) -- gratefully credited. Specific sources:
L. P. Jorge & W. H. Meeks III (1983) k-noids; C. J. Costa (1982),
D. Hoffman & W. H. Meeks III (1985+) Costa-Hoffman-Meeks; H. Karcher (1988) saddle towers ("Embedded minimal surfaces derived from Scherk's examples", Manuscripta Math. 62);
W. H. Meeks III (1981) the minimal Mobius strip ("The classification of complete minimal surfaces in R3 with total curvature greater than -8pi", Duke Math. J. 48);
B. Riemann (1867) the singly periodic minimal example foliated by circles; H. A. Schwarz (1890) the Bjorling formula;