Tangent Developable

Tangent Developable is a ruled surface, given by a parametrisation, immersed.

Open Tangent Developable in the interactive viewer →

aperiodic def-parametric immersed implemented ruled

Formula

x(u, v)

\cos\left(u\right) - v \sin\left(u\right)

y(u, v)

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

z(u, v)

0.4 \left(u + v\right)

Properties

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

Definition

tangent developable of the unit helix of pitch 0.4 over 2 turns (operator defaults); v starts at the 0.04 edge gap off the cuspidal edge of regression (the helix itself)

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.28 of tol, 0.1% of oracle beyond tol (tol floor 2.5 x 0.04382).

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.