Twisted Torus (Parametric)

Questions
1

I’m trying to replicate the “twisted torus” displayed here: Chromatic ice cube 🌈🧊 : generative (not the moving cube, just the main rotating torus shape)

I got close with the Gray’s Klein bottle parametric equations here: Toroidal Nature

I think the main difference is that Gray’s klein bottle sweeps a figure 8 around a circular path, whereas the Reddit reference link looks like it sweeps a square or something, but I’m not sure how to map that difference to parametric equations. Any help is much appreciated!

Here’s the shape I have so far, notice the “ribs” which I am trying to get rid of (they are caused by the figure 8 mentioned above):

klein-2
klein-21549×1502 236 KB

Here is my code for the geometry of the above:

const surface = (u: number, v: number, target: Vector3) => {
  const a = 4,
    n = 3,
    m = 2;

  const i = (n * u) / 2,
    j = (m * u) / 2,
    k = Math.sin(2 * v);

  const x = (a + Math.cos(i) * Math.sin(v) - Math.sin(i) * k) * Math.cos(j),
    y = (a + Math.cos(i) * Math.sin(v) - Math.sin(i) * k) * Math.sin(j),
    z = Math.sin(i) * Math.sin(v) + Math.cos(i) * k;

  target.set(x, y, z);
};

const geometry = new ParametricGeometry(surface, 100, 100);
2

Is this relatively close to what you need:

https://codepen.io/boytchev/full/qBLKpjx

image

There are some options for modifications:

  • change line 46: n=10 → n=50 to get sharper edges
  • change line 47: t=1.5 → t=5 to get more twists

image
image1431×708 42 KB

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3

@PavelBoytchev that is EXACTLY what I’m looking for. Thank you so much, you’ve made my day! :slight_smile:

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4

pretty ~~~

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5

How did you derive the parametric equations?

6

This is called Math.

I started with a torus. Its parametric equation does two circles – one big circle along the whole torus (using u as parametric angle) and one small circle across the tube (using v as parametric angle).

x = (R+r*cos(v)) * Math.cos(u)
y = (R+r*cos(v)) * Math.sin(u)
z = r*sin(v)

image

Then I changed the radius r of the small circle so that it is not a circle, but a superellipse. A super ellipse can make a smooth transition from a circle to a square, so instead of a constant radius r, I used:

r = (cos(v)**n + sin(v)**n)**(-1/n)

image

Finally, I added the twist – this is u and v are angles that go along the two circles, and I made one of the angle to be added to the other angle by a factor t, so instead of cos(v) and sin(v) I used:

cos(v+t*u)
sin(v+t*u)

image

Thus the overall equation looks like this (R=4):

r = (cos(v)**n + sin(v)**n)**(-1/n)
x = (4 + r*cos(v+t*u)) * cos(u)
y = (4 + r*cos(v+t*u)) * sin(u)
z = r*sin(v+t*u)

You can get all the shapes above by simply changing t and n:

  • torus with circular tube: t=0, n=2
  • torus with square tube: t=0, n=10
  • square torus with a twist: t=1.5, n=10
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