Bryant Surface
Bryant Surface is a constant-mean-curvature surface, of constant mean curvature one in the Bryant sense, given by a parametrisation, immersed.
Open Bryant Surface in the interactive viewer →
aperiodic cmc cmc1-bryant def-parametric immersed implemented
Properties
- Family
- cmc
- Given by
- a parametrisation
- Curvature
- constant mean curvature one in the Bryant sense
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
Definition
No chart is stored: the CMC-1 surface is built by numerically integrating Bryant's holomorphic representation in H^3 and then mapping through the Poincare-ball model; there is no closed real chart to transcribe. 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.
- Robert L. Bryant, "Surfaces of mean curvature one in hyperbolic space", Asterisque 154-155 (1987), 321-347. Theorem A (the representation f = F (conj F)^T), Theorem B (the polynomial data on C), Example 1 (Enneper's cousin, p. 341), Example 2 (the catenoid cousins, pp. 341-342).
- A. I. Bobenko, T. V. Pavlyukevich, B. A. Springborn, "Hyperbolic constant mean curvature one surfaces: spinor representation and trinoids in hypergeometric functions", Math. Z. 245 (2003), 63-91; arXiv:math/0206021 -- the spinor form of the same representation and the explicit trinoids the TRINOID mode implements (see math_art/trinoid.py for the construction, gates and details).
- M. Umehara and K. Yamada, "Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space", Ann. of Math. 137 (1993), 611-638 -- the global theory these examples opened up.