Tannery's Pear

Tannery's Pear is a surface of revolution, given by a parametrisation, genus 0, immersed with singularities.

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aperiodic closed closed-orientable def-parametric implemented revolution singular tradition-classical

Formula

x(u, v)

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

y(u, v)

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

z(u, v)

\sqrt{8} \sin\left(u\right)

Properties

Family: revolution
Given by: a parametrisation
Curvature: no condition imposed
Periodicity: not periodic
Genus: 0
Ends: 0
Embedding: immersed with singularities
Orientable: yes
Fidelity: exact
Exactness: elementary

Definition

half a Gerono lemniscate (rho = sin u cos u) stretched vertically by 2 sqrt2 and revolved, at the operator's default scale a = 1 -- the default TANNERY_PEAR variant (the hourglass revolves the whole lemniscate, u in [-pi/2, pi/2]). The builder fits its output to the 2 m cube; verification applies the same centre-and-scale to the chart

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.67 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.01537).

J. Tannery (1892); the surface of revolution of half a Gerono lemniscate.