Spiral Ruled

Spiral Ruled is a ruled surface, given by a parametrisation, immersed.

Open Spiral Ruled in the interactive viewer →

aperiodic def-parametric immersed implemented ruled

Formula

x(u, v)

\exp\left(0.15 u\right) \left(\cos\left(u\right) + v \left(0.15 \cos\left(u\right) - \sin\left(u\right)\right)\right)

y(u, v)

\exp\left(0.15 u\right) \left(\sin\left(u\right) + v \left(0.15 \sin\left(u\right) + \cos\left(u\right)\right)\right)

z(u, v)

v

Properties

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

Definition

Farris' spiral ruled surface at the operator defaults: log-spiral base of tightness 0.15 (plain spiral, no rosette), 2 turns, ruling slope 1 and half-length 1

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.33 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.1462).

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).
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".
Milk carton: H. M. Cundy & A. P. Rollett, "Mathematical Models" (1951), 185-188.