Math Art
Dupin Cyclide

Dupin Cyclide

Dupin Cyclide is a cyclide, given by a parametrisation, genus 1, embedded or immersed depending on parameters.

Open Dupin Cyclide in the interactive viewer →

aperiodic closed closed-orientable cyclide def-parametric implemented tradition-classical tradition-physical-model

Formula

x(u, v)
1.9+((1+0.45⁢cos⁡(v))⁢cos⁡(u)−1.9)⁢2.56((1+0.45⁢cos⁡(v))⁢cos⁡(u)−1.9)2+((1+0.45⁢cos⁡(v))⁢sin⁡(u))2+(0.45⁢sin⁡(v))21.9 + \frac{\left(\left(1 + 0.45 \cos\left(v\right)\right) \cos\left(u\right) - 1.9\right) \cdot 2.56}{\left(\left(1 + 0.45 \cos\left(v\right)\right) \cos\left(u\right) - 1.9\right)^{2} + \left(\left(1 + 0.45 \cos\left(v\right)\right) \sin\left(u\right)\right)^{2} + \left(0.45 \sin\left(v\right)\right)^{2}}
y(u, v)
(1+0.45⁢cos⁡(v))⁢sin⁡(u)⁢2.56((1+0.45⁢cos⁡(v))⁢cos⁡(u)−1.9)2+((1+0.45⁢cos⁡(v))⁢sin⁡(u))2+(0.45⁢sin⁡(v))2\frac{\left(1 + 0.45 \cos\left(v\right)\right) \sin\left(u\right) \cdot 2.56}{\left(\left(1 + 0.45 \cos\left(v\right)\right) \cos\left(u\right) - 1.9\right)^{2} + \left(\left(1 + 0.45 \cos\left(v\right)\right) \sin\left(u\right)\right)^{2} + \left(0.45 \sin\left(v\right)\right)^{2}}
z(u, v)
0.45⁢sin⁡(v)⁢2.56((1+0.45⁢cos⁡(v))⁢cos⁡(u)−1.9)2+((1+0.45⁢cos⁡(v))⁢sin⁡(u))2+(0.45⁢sin⁡(v))2\frac{0.45 \sin\left(v\right) \cdot 2.56}{\left(\left(1 + 0.45 \cos\left(v\right)\right) \cos\left(u\right) - 1.9\right)^{2} + \left(\left(1 + 0.45 \cos\left(v\right)\right) \sin\left(u\right)\right)^{2} + \left(0.45 \sin\left(v\right)\right)^{2}}

Properties

Family
cyclide
Given by
a parametrisation
Curvature
no condition imposed
Periodicity
not periodic
Genus
1
Ends
0
Embedding
embedded or immersed depending on parameters
Orientable
yes
Fidelity
exact
Exactness
elementary

Definition

the record's default variant, the RING cyclide: the ring torus R = 1, r = 0.45 inverted in the sphere of radius 1.6 about (1.9, 0, 0) -- all operator defaults. Inversion carries the torus's circular curvature lines to circles, which is the property that defines a Dupin cyclide

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.30 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.02719).