Helicoidal Cone

Helicoidal Cone is a ruled surface, given by a parametrisation, immersed.

aperiodic def-parametric immersed implemented ruled tradition-classical

Formula

x(u, v)

u \cos\left(v\right)

y(u, v)

u \sin\left(v\right)

z(u, v)

0.5 u v

the cone joining the origin to the helix (cos v, sin v, 0.5 v), two turns -- k = amp = 0.5 and turns = 2, the operator defaults

Chart reproduced numerically against the shipped implementation: fwd max 0.70 of tol, 0.1% of oracle beyond tol (tol floor 2.5 x 0.06159).

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.
R. Ferreol, "Helicoidal cone (Cone helicoidal)", Encyclopedie des formes mathematiques remarquables, mathcurve.com; local mirror chapter ch1064_conehelicoidal (S:/data/math_art/references/websites/mathcurve/book/surfaces/ch1064_conehelicoidal.md).