The textbook local image warp folds near its own rim, and worst at small displacements

Fact

Gustafsson's local translation — the primitive behind every "liquify" brush and face-slimming filter — maps a pixel back with

u = x - ((r² - d²) / (r² - d² + |m|²))² · m        d = |x - c|

That ratio is flat in the middle and steep at the rim: it climbs from 0 to nearly 1 over a band only about |m|²/2r wide. Sampling the axial backward map numerically, it is non-monotonic — folded — for every |m| below about 0.55 r, and the fold peaks at small displacements:

|m|/r    0.10   0.20   0.30   0.45   0.50   0.55
fold      .044   .052   .041   .013   .005   0      (in units of r)

So the intuitive rule "keep the displacement well under the radius" is backwards for this falloff. Replacing it with a falloff that is flat at both ends, 1 - t²(3 - 2t) with t = d/r, folds nowhere below |m| = 2r/3, and its handle has an exact fixed point instead of an approximate one.

Why it matters

The fold is a thin ring inside each handle where a few percent of r worth of source content is sampled twice. It renders as a faint crease or a ripple, and because every handle is individually correct and the image looks fine at high strength, it reads as a bad mask, a bad mesh, or camera noise. Turning the effect down to investigate makes it worse.

How to apply

• Use a falloff with zero slope at the rim — 1 - t²(3 - 2t), a raised cosine, anything C¹ — for both translation and radial-scale handles. The same property is what lets primitives sum into a field with no creases ([[raised-cosine-contours-compose-without-seams]]).
• Derive the safe cap from the falloff's own steepest slope: injective while |m| · max|S'| < r. For 1 - t²(3 - 2t) that maximum is 1.5, hence 2r/3.
• Centre a translation handle on the destination; then u(c) = c - m exactly, which is also a one-line assertion in a test.
• Assert the fold threshold in a test rather than trusting the cap: walk the axial backward map and require a strictly positive slope.