Parabolic Conoid

Parabolic Conoid is a ruled surface, given by a parametrisation, immersed.

aperiodic def-parametric immersed implemented ruled tradition-architectural tradition-classical

Formula

x(u, v)

v

y(u, v)

u

z(u, v)

0.5 v \left(1 - u^{2}\right)

the roof-shell conoid a^2 z = x(b^2 - y^2) with b = 1 and amp = 1/a^2 = 0.5, ruling extent 1 -- the operator defaults

Chart reproduced numerically against the shipped implementation: fwd max 0.27 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.02813).

Parabolic conoid (a2 z = x(b2 - y2), the roof-shell conoid with a parabola directrix), sinusoidal cone (cylindrical equation z = k rho cos(n theta), the cone over the sine-wave crown) and helicoidal cone (x = a u cos v, y = a u sin v, z = b u v, the cone joining an apex to a circular helix): R. Ferreol, ibid., chapters "conoide parabolique", "cone sinusoidal" and "cone helicoidal".
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".
R. Ferreol, "Parabolic conoid", Encyclopedie des formes mathematiques remarquables, mathcurve.com; local mirror chapter ch1240_conoide_parabolique_2 (S:/data/math_art/references/websites/mathcurve/book/surfaces/ch1240_conoide_parabolique_2.md).