Revolution of the Sinusoid

Revolution of the Sinusoid is a surface of revolution, given by a parametrisation, immersed.

Open Revolution of the Sinusoid in the interactive viewer →

aperiodic def-parametric immersed implemented revolution tradition-classical

Formula

x(u, v)

\cos\left(u\right) \cos\left(v\right)

y(u, v)

\cos\left(u\right) \sin\left(v\right)

z(u, v)

u

Properties

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

Definition

x = a cos(z/b) about Oz with a = b = 1 and three beads, the operator defaults; the signed radius only shifts the revolution angle by pi, so the chart and the welded-bead mesh cover the same point set

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.31 of tol, 1.4% of oracle beyond tol (tol floor 2.5 x 0.04627).

Revolution of the catenary (alysseid), z = a cosh(rho/a) about the catenary's axis of symmetry -- the catenoid's companion, and not minimal: R. Ferreol, ibid., chapter "alysseide".
Revolution of the sinusoid, x = a cos(z/b) about Oz -- a string of onion-dome beads meeting in cusp circles on the axis: R. Ferreol, ibid., chapter "surface de revolution de la sinusoide" (the chapter dates its study to 2012).
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;
R. Ferreol, "Revolution of the sinusoid", Encyclopedie des formes mathematiques remarquables, mathcurve.com; local mirror chapter ch1328_revolsin_2 (S:/data/math_art/references/websites/mathcurve/book/surfaces/ch1328_revolsin_2.md).