Elastic Torus (S3)

Properties

Family

cmc

Given by

a parametrisation

Curvature

no condition imposed

Periodicity

not periodic

Ends

0

Embedding

immersed

Fidelity

exact

Definition

No chart is stored: the generating elastica is solved numerically (elliptic functions / shooting) before the Hopf lift; outside the language. The shipped implementation is authoritative; an unverified transcription would silently define a different surface.

Sources

Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.

• Charles-Eugene Delaunay, "Sur la surface de revolution dont la courbure moyenne est constante", J. Math. Pures Appl. 6 (1841), 309-314 (the classification and the rolling-conic theorem).
• M. Sturm, note appended to Delaunay's paper (1841), 315-320 -- the first integral used here.
• James Eells, "The surfaces of Delaunay", Math. Intelligencer 9 (1987), 53-57 (a modern account of the roulette construction).
• Nicholas J. Korevaar, Rob Kusner, Bruce Solomon, "The structure of complete embedded surfaces with constant mean curvature", J. Diff. Geom. 30 (1989), 465-503 -- Delaunay surfaces as the ends of every complete embedded CMC surface, which is why this family matters beyond its own good looks.
• Lynn Heller, "Constrained Willmore tori and elastic curves in 2-dimensional space forms", Comm. Anal. Geom. 22 (2014), no. 2; arXiv:1303.1445 -- the ELASTIC_TORUS mode: elastic curves in H^2 in Weierstrass closed form (Thm. 2), their closure (Thm. 4), and the CMC-in-S^3 classification of the revolved tori (Prop. 6).
• Joel Langer, David A. Singer, "Curves in the hyperbolic plane and mean curvature of tori in 3-space", Bull. London Math. Soc. 16 (1984), 531-534 -- a torus of revolution is Willmore-critical exactly when its profile curve is elastic in the hyperbolic plane.