Scherk's Doubly Periodic Surface
Scherk's Doubly Periodic Surface is a doubly periodic minimal surface, given by a Weierstrass representation, genus 0, 4 ends, embedded, with a layer group.
Open Scherk's Doubly Periodic Surface in the interactive viewer →
def-weierstrass doubly-periodic embedded implemented minimal minimal-periodic tradition-classical
Formula
Properties
- Family
- minimal-periodic
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- doubly periodic
- Genus
- 0
- Ends
- 4
- Embedding
- embedded
- Symmetry kind
- layer group
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
Extracted from the shipped row's source by AST (complete multi-line lambdas, the shipped tilted-Scherk forms (_tiltscherk_phi) at Rho = 1, which the zoo states reproduces SCHERK1 exactly) and reproduced numerically against the shipped callables over the complex plane, exactly -- no scalar slack.
Sources
Weierstrass data reproduced numerically against the shipped implementation (math_art/minsurf/zoo.py, row SCHERK1) by sampling g and dh over rings in the complex plane at the row's default parameters: g: matches over 144 complex samples (worst 4e-16); dh: matches over 144 complex samples (worst 0).
- H. F. Scherk, 'Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen', J. reine angew. Math. 13 (1835).
- H. F. Scherk, 'Bemerkungen ueber die kleinste Flaeche innerhalb gegebener Grenzen', J. Reine Angew. Math. 13 (1835) 185-208.
- H. Lazard-Holly, W. H. Meeks III, 'Classification of doubly-periodic minimal surfaces of genus zero', Invent. Math. 143 (2001) 1-27; M. Weber (mirror: minsurf ch046): 'The quotient surface by its translational symmetries is a 4-punctured sphere.'