Oloid

Oloid is a ruled surface, of zero Gaussian curvature, given by a parametrisation, genus 0, embedded.

aperiodic closed closed-orientable def-parametric embedded flat implemented ruled tradition-classical tradition-sculptural

Formula

x(u, v)

\left(1 - v\right) \sin\left(u\right)

y(u, v)

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

z(u, v)

\frac{v \sqrt{1 + 2 \cos\left(u\right)}}{1 + \cos\left(u\right)}

Properties

Family: ruled
Given by: a parametrisation
Curvature: zero Gaussian curvature
Periodicity: not periodic
Genus: 0
Ends: 0
Embedding: embedded
Orientable: yes
Fidelity: exact
Exactness: elementary

Sources

Chart as curated in tools/surfdb/charts.py, verified numerically against the curvature condition the record claims (measured over the chart by the validator and the charts self-test).

P. Schatz, Rhythmusforschung und Technik (1975); the oloid dates from his 1929 work on the invertible cube.
H. Dirnboeck, H. Stachel, 'The development of the oloid', J. Geom. Graphics 1 (1997) 105-118 -- area 4*pi*r^2.