Twisted Torus
Shape · (x, y, z) = f(u, v, t)
((3 + cos(v + 2u + t)) · cos(u), sin(v + 2u + t), (3 + cos(v + 2u + t)) · sin(u))
Open in the app The dials and keys named below are the app's.
What it draws
This is a torus of centre radius 3 and tube radius 1, written with the same angle v + 2u + t in all three expressions. The bracket 3 + cos(v + 2u + t) is the distance from the vertical axis, and the width and depth carry it round as cos(u) and sin(u). The second expression is the height, sin(v + 2u + t).
The same donut
Adding 2u + t to v only shifts where the tube section starts. As v sweeps a full 2π it covers the same circle whatever the shift is, so the surface is exactly the plain torus. Every point on screen is still 1 away from the centre circle of radius 3, at every instant. The outer edge is at 4, the hole at 2, and the tube reaches ±1 in height.
What the twist changes is the grid, not the shape. The mesh lines are lines of constant v, and the 2u makes such a line wind twice round the tube while it goes once round the ring. It closes back on itself after a single lap, a rope laid on the donut rather than a hoop.
The twist in time
The t is added to every section at once, so all of them turn together at one radian per second at Speed 1. The ribbing appears to crawl along the tube and comes back to where it began after 2π ≈ 6.28 seconds, while the surface underneath never moves at all. It is the clearest case in the app of a moving parametrisation over a still object.
Try
- Push Mesh to its top stop. The lit solid is an ordinary donut, and the animation vanishes with the grid that carried it.
- Change every 2u to 3u, as in (3 + cos(v + 3u + t))·cos(u): three windings of the tube per lap instead of two.
- Drop the t from all three expressions and the twist freezes, leaving a static braid.
- Drop the 2u instead, keeping v + t: the sections all point the same way again and the whole grid rolls round the tube in step.
- Top in the deck: from above the outline is the same pair of circles as the plain torus.