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Hopf fibration fibres as linked circles

In your harness4D geometry toolMIT

Lift base points of S2 to their Hopf fibres in S3, project them stereographically into linked circles, check the fibration's properties (great circles, Clifford parallels, linking number 1, tori over latitude circles), and write tubes plus a colour-matched base sphere as glTF.

Render of Hopf fibration fibres as linked circles
The render its check recorded in Godot 4.7.2, 256 by 256 pixels.

What it takes and gives

Entry points
hopf_map, lift, fibre, base_points, colour, is_great_circle, is_circle_3d, linking_number and 6 more
Outputs
example.gltf
Coordinates
quaternion a + bi + cj + dk at (x, y, z, w) = (b, c, d, a); stereographic image from (0, 0, 0, 1), Godot/glTF +Y up

Limits. Fibres are polygons of the given sample count, so their images are polygonal circles. Base circles must avoid the x axis direction, whose fibre passes through the projection pole and becomes a line; fibres near it become very large. The inset base sphere uses alpha blending.

Details

Kind
4D geometry tool
Engine
Python 3.10 or later
Files
10 files, 328.6 KB
Licence
MIT
Declared effects
reads files, runs commands, writes files
Checked with
Godot 4.7.2, gl_compatibility (opengl3, Mesa llvmpipe under Xvfb)
Checks
cli rebuilds example, gltf structure example, cli example runs, cli example gltf structure, godot loads example, godot loads cli clifford fibres, preview render
Techniques
Hopf fibration S3 to S2, Hopf map q i conj(q), Clifford parallel great circles, Hopf link (linking number 1), stereographic projection, Gauss linking integral, parallel transport frames for swept tubes
Tags
4d, hopf fibration, 3-sphere, linked circles, villarceau circles, clifford parallels, stereographic projection, gltf
Identity
library.supply.creative_originals.8cfeb9c3b317baca766adef7.03f754d58667be59
Version
03f754d58667be59d73de15e18afdc2fa93a23882c4d2d9fe4876efd35bfd03d

The version is the SHA-256 of the item your harness downloads; every file in it names its own checksum.

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