Hyperboloid of One Sheet

Hyperboloid of One Sheet is a quadric surface, given by a parametrisation, genus 0, embedded, with a continuous symmetry group.

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aperiodic def-parametric embedded implemented noncompact quadric tradition-architectural tradition-classical

Formula

x(u, v)

\left(1 - v\right) \cos\left(u\right) + v \cos\left(u + \frac{2 \pi}{3}\right)

y(u, v)

\left(1 - v\right) \sin\left(u\right) + v \sin\left(u + \frac{2 \pi}{3}\right)

z(u, v)

2 v - 1

Properties

Family: quadric
Given by: a parametrisation
Curvature: no condition imposed
Periodicity: not periodic
Genus: 0
Ends: 0
Embedding: embedded
Orientable: yes
Symmetry kind: continuous symmetry group
Fidelity: exact
Exactness: elementary

Definition

the stick hyperboloid: straight rulings between two unit circles at z = -1 and z = +1, the top rotated by the operator's default 120-degree twist (waist radius cos 60 = 1/2)

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 0.04328).

Classical; one of the two doubly ruled quadrics.