Paper Bag Surface

Paper Bag Surface is a surface, given by a parametrisation, immersed.

Open Paper Bag Surface in the interactive viewer →

aperiodic def-parametric immersed implemented misc

Formula

x(u, v)

v \cos\left(u\right)

y(u, v)

\left(v - 1.26 u\right) \sin\left(u\right)

z(u, v)

2.47 v^{2}

Properties

Family: misc
Given by: a parametrisation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed
Fidelity: exact
Exactness: elementary

Definition

the classic constants a = 2.47 (height) and b = -1.26 (crimp) and depth 2 fixed to the operator defaults; the u = 0 and u = 2*pi boundary curves coincide, so u wraps

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.69 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.09735).

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.