Proportional-Curvature Surface
Proportional-Curvature Surface is a surface of revolution, a Weingarten surface, given by a parametrisation, immersed.
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aperiodic def-parametric immersed implemented revolution tradition-classical weingarten
Properties
- Family
- revolution
- Given by
- a parametrisation
- Curvature
- Weingarten surface
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
Definition
The surfaces of revolution whose principal curvatures are in a constant ratio k: meridian rho = a cos^k(t). k = 1 gives the sphere, k = -1 a minimal surface (catenoid); the four sign/size regimes of k give spindles, domes, trumpets and annular waists. SHIPPED with k a live knob and the defining relation itself measured: the self-test computes the discrete meridian/parallel curvature ratio on the produced meridian and checks it equals k, converging under refinement, for k in {1, 2, -1, 0.5}; the k = 1 member is verified to be the round sphere and k = -1 exactly the catenoid rho = a cosh(z/a).
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- D. von Seggern, "CRC Standard Curves and Surfaces" (1993), p. 304 -- Fresnel's elasticity surface.
- A. Robin (2004), plotted after M. Trott, "The Mathematica GuideBook for Graphics" (2004), p. 103 -- the paper bag surface.
- O. Knill (2017); M. A. Villarino and J. C. Varilly (2024) -- the trihyperboloid and its volume ln 256.
- C. Dupin, "Applications de Geometrie et de Mechanique", Paris 1822;
- B. Odehnal, "Ortho-Circles of Dupin Cyclides", J. Geometry Graphics 10 (2006) 1-21 -- the inversion construction and the ring / horn / spindle classification.
- J. Tannery, Bulletin des sciences mathematiques, 2e serie, 16 (1892) 190; O. Zoll, "Ueber Flaechen mit Scharen geschlossener geodaetischer Linien", Math. Ann. 57 (1903) 108-133; A. L. Besse, "Manifolds all of whose Geodesics are Closed", Springer 1978.
- R. Ferreol, "Rotation surface with proportional curvatures", Encyclopedie des formes mathematiques remarquables, mathcurve.com; local mirror chapter ch1365_revolpropor_2 (S:/data/math_art/references/websites/mathcurve/book/surfaces/ch1365_revolpropor_2.md).
- W. Kuhnel, "Differential Geometry: Curves - Surfaces - Manifolds", p. 95, where the family is worked (the chapter's own reference, dated 1999).