Generalized ellipse windings

One equation, nine parameter presets. Select a panel to edit it; drag any drawing to rotate its view. These are illustrative approximations, not recovered equations for the reference image.

Equation and interpretation

Each separate loop is an exact ellipse in 3D:

p(t, φ) = Ry(α(φ)) Rz(β(φ)) Ry(γ(φ)) [b sin(t), a cos(t), 0]ᵀ
α(φ) = Pφ
β(φ) = β₀ + A sin(mφ + δ)
γ(φ) = Kφ + B sin(nφ)
0 ≤ t < 2π,  0 ≤ φ < 2π

Rotations act from right to left: local twist, axis tilt, then precession. The camera applies another rotation and orthographic projection. Varying φ between closed loops gives a family of ellipses. In continuous mode, t=Nφ: orientation changes during each turn, so the strand is generally not an ellipse. For integer P, K, m, n, and N, that strand closes after one full φ sweep.

Orthographic projection maps each planar ellipse to an ellipse or, in an edge-on view, a line segment. Apparent crossings do not imply that strands meet in 3D.