This is a genuinely two-dimensional scattering model — not a flattened view of a 3D structure. The bead chain itself lives in a plane (e.g. a molecule confined to an interface, or a 2D cross-section ensemble), and it tumbles by rotating within that plane, not in 3D. Averaging a scattering vector's phase over all in-plane rotation angles gives a different exact closed form than the 3D case: instead of the sinc function, the orientation-averaged pair contribution is the Bessel function J₀:
I(q) = Σᵢ Σⱼ J₀(q·rᵢⱼ) rᵢⱼ = |rᵢ − rⱼ| (2D distance)
⟨cos(q·r)⟩ averaged over a 2D rotation angle is exactly J₀(qr) — the 2D analogue of the sinc(qr) that appears for a 3D orientation average. J₀ is computed here from its own polynomial/asymptotic series, not looked up.
The Guinier approximation also picks up a different constant in 2D. Expanding the exact scattering function for small q, the projection of R_g² onto any one of d spatial dimensions is R_g²/d, so:
I(q) ≈ I(0)·exp(−q²Rg²/2) (2D; the 3D version uses /3) valid for q·Rg < 1.3
so fitting ln I(q) against q² over the low-q region recovers Rg from slope = −Rg²/2, compared here against the direct real-space value:
Rg² = (1/N) Σᵢ |rᵢ − r_cm|² (2D coordinates)
- Fold shape — Globular packs beads uniformly inside a disk of area ∝ N (2D compact scaling, Rfold ∝ √N); Elongated stretches that disk into an ellipse.
- Denaturation slider — morphs the beads toward an extended 2D random walk. Rg grows, I(0) stays N² (same total scattering mass), and the Guinier region shrinks — the same diagnostic a structural biologist reads off real SAXS data.
- Chain length N — more residues, more scattering pairs, a bigger molecule.
- Thermal motion — small in-plane jitter mimicking solution-phase fluctuation; perturbs Rg/Dmax frame to frame.
- Show all pair-lines — draws every i–j segment used in the Debye double sum, so you can see exactly what is being summed.
Dmax, the largest pairwise distance in the bead cloud, is drawn as a dashed segment on the canvas.