Four-Soliton Surface
Four-Soliton Surface is a surface of constant negative Gaussian curvature, given by a parametrisation, immersed.
Open Four-Soliton Surface in the interactive viewer →
aperiodic constant-curvature def-parametric immersed implemented k-const-negative
Properties
- Family
- constant-curvature
- Given by
- a parametrisation
- Curvature
- constant negative Gaussian curvature
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
Definition
No chart is stored: the multi-soliton sine-Gordon surface is mapped onto a lab window (`_soliton_window`) FITTED numerically from the soliton speeds and charges, so even though the multi-soliton formula itself is explicit, the domain the operator draws is not a stated u/v rectangle and a chart would misreport the framing. 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.
- Sym's formula: A. Sym, "Soliton surfaces and their application", in: Soliton geometry from spectral problems, Lecture Notes in Physics 239, Springer, Berlin 1985, 154-231 -- the immersion as the logarithmic derivative of the frame in the spectral parameter, F = 2 rho Phi^{-1} dPhi/dt; used here exactly as stated in Bobenko 1994, Theorem 11.
- Pseudosphere: Eugenio Beltrami, "Saggio di interpretazione della geometria non-euclidea", Giornale di Matematiche 6, 1868.
- Dini's surface: Ulisse Dini, 1865.
- Kuen's surface: Theodor Kuen, "Ueber Flaechen von constantem Kruemmungsmass", Sitzungsber. Bayer. Akad. Wiss., 1884.
- Hilbert's theorem: D. Hilbert, "Ueber Flaechen von constanter Gausscher Kruemmung", Trans. AMS 2, 1901, pp. 87-99.
- Minding bulge and spindle: Ferdinand Minding, "Wie sich entscheiden laesst, ob zwei gegebene krumme Flaechen auf einander abwickelbar sind oder nicht; nebst Bemerkungen ueber die Flaechen von unveraenderlichem Kruemmungsmasse", J. reine angew. Math. (Crelle) 19 (1839), 370-387 -- the classification of the constant-curvature surfaces of revolution into the parabolic, hyperbolic and conic types, and the theorem that all surfaces of equal constant curvature are locally isometric.