Compound Helical Cone

Compound Helical Cone is a ruled surface, given by a parametrisation, immersed.

Open Compound Helical Cone in the interactive viewer →

aperiodic def-parametric immersed implemented ruled

Formula

x(u, v)

\left(1 - v\right) \left(1 + 0.18 \left(1 - v\right) \cos\left(6 u + 4 \pi v\right)\right) \cos\left(u\right)

y(u, v)

\left(1 - v\right) \left(1 + 0.18 \left(1 - v\right) \cos\left(6 u + 4 \pi v\right)\right) \sin\left(u\right)

z(u, v)

2.5 v

Properties

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

Definition

base radius 1, height 2.5, 6 flutes of depth 0.18 winding 2 turns, straight taper, no planetary orbit -- the operator defaults. The flutes fade linearly to arrises at the apex v = 1

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.28 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.03437).

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.