Hopf Torus

Hopf Torus is a topological surface, given by a parametrisation, immersed.

Open Hopf Torus in the interactive viewer →

aperiodic def-parametric immersed implemented topological

Formula

x(u, v)

\frac{\cos\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \cos\left(3 u\right)\right) \cos\left(u + v\right)}{1 - \sin\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \cos\left(3 u\right)\right) \sin\left(v\right)}

y(u, v)

\frac{\cos\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \cos\left(3 u\right)\right) \sin\left(u + v\right)}{1 - \sin\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \cos\left(3 u\right)\right) \sin\left(v\right)}

z(u, v)

\frac{\sin\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \cos\left(3 u\right)\right) \cos\left(v\right)}{1 - \sin\left(\frac{\pi}{4} + \frac{17.5 \pi}{180} \cos\left(3 u\right)\right) \sin\left(v\right)}

Properties

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

Definition

the operator's default WAVY curve: colatitude beta(u) = pi/2 + (35 deg) cos 3u on S^2, lifted to its Hopf fibre torus in S^3 and stereographically projected from (0,0,0,1). The builder centres on the bounding box and scales the 97th-percentile radius to 1; verification applies the same fit to the chart

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.52 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.007408).

Heinz Hopf, "Ueber die Abbildungen der dreidimensionalen Sphaere auf die Kugelflaeche", Math. Ann. 104 (1931), 637-665 (the fibration and its linking invariant).
D. W. Lyons, "An Elementary Introduction to the Hopf Fibration", Math. Mag. 76 (2003), 87-98.
N. Johnson, "Visualization of the Hopf fibration" (2011), https://nilesjohnson.net/hopf.html (the colour/latitude-torus rendering imitated here).
Y. Villarceau (1848): a torus of revolution carries two extra circles through each point, the "Villarceau circles" -- exactly the Hopf fibres of a stereographically projected Clifford torus.
Ulrich Pinkall, "Hopf tori in S^3", Invent. Math. 81 (1985), 379-386 (the Hopf torus over a curve on S^2, and the theorem that it is Willmore exactly over an elastic curve).
Luigi Bianchi (1894): classification of the flat surfaces immersed in S^3; the case with one family of asymptotic lines a great circle is exactly the Hopf tori. Modern treatment in M. Spivak, "A Comprehensive Introduction to Differential Geometry", vol. IV, p. 139ff.