Catalan's Minimal Surface
Catalan's Minimal Surface is a minimal surface, given by a Weierstrass representation, self-intersecting.
Open Catalan's Minimal Surface in the interactive viewer →
aperiodic def-weierstrass implemented minimal self-intersecting tradition-classical
About
The minimal surface that contains a cycloid as a geodesic -- the curve traced by a point on a rolling wheel. Catalan found it in 1855 by asking which minimal surface a given curve could sit inside, which is the question Björling's problem answers in general.
Properties
- Family
- minimal
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- self-intersecting
- Orientable
- yes
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
Bjorling's problem: the unique minimal surface through a given real-analytic strip. The defining seed, reproduced numerically against the shipped row (math_art/minsurf/zoo.py, BJ_CYCLOID): c(t) = (t - sin(t), 1.0 - cos(t), 0), with normal n(t) = (cos(t/2), -sin(t/2), 0), t in [-3.141592654, 9.424777961]. The Weierstrass data is the holomorphic extension of the seed, computed numerically by the engine -- the seed, not a (g, dh) pair, is what specifies this surface.
Sources
Bjorling seed reproduced numerically against the shipped implementation (math_art/minsurf/zoo.py, row BJ_CYCLOID): seed reproduced against the shipped callables over 200 samples (worst 0).
- E. Catalan, 'Memoire sur les surfaces dont les rayons de courbure en chaque point sont egaux et de signes contraires', C. R. Acad. Sci. Paris 41 (1855).
- E. Schwarz's Bjorling construction realises it from a cycloid.