Guimard's Surface

Guimard's Surface is a ruled surface, given by a parametrisation, immersed.

Open Guimard's Surface in the interactive viewer →

aperiodic def-parametric immersed implemented ruled

Formula

x(u, v)

\cos\left(u\right)

y(u, v)

v \sin\left(u\right)

z(u, v)

0.5 v \sin\left(u\right)^{2}

Properties

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

Definition

the segment from M = (cos u, 0, 0) on a line to N = (cos u, sin u, 0.5 sin^2 u) on a circle (operator defaults a = b = 1, c = 0.5); both points share their x, so the rulings run in planes x = const

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.27 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.04688).

Antoni Gaudi (1852-1926), the ruled roof of the Sagrada Familia escoles; Hector Guimard (1867-1942). Both parametrizations, and the milk carton's, from R. Ferreol, "Encyclopedie des formes mathematiques remarquables" (mathcurve.com), chapters "surface de Gaudi", "surface de Guimard" and "berlingot".
G. W. Hart, "Curved, yet Straight: Stick Hyperboloids," Bridges 2023 Conference Proceedings, pp. 251-258.
E. Jannasch & J. Macnab, "The Compound Helical Cone as Kinematic Trace," Bridges 2023 Conference Proceedings, pp. 15-22.
F. A. Farris, "Spiral Ruled Surfaces," Bridges 2022 Conference
Proceedings, pp. 289-292; and F. A. Farris, "Creating Symmetry" (Princeton Univ. Press, 2015).
Milk carton: H. M. Cundy & A. P. Rollett, "Mathematical Models" (1951), 185-188.