Pinch

Pinching the paper folds the water toward one axis and stretches it along the other. In pinch-local coordinates (rotated by the fold angle about the pinch centre):

$$x_{src} = x\,e^{+k\,w(s)},\qquad y_{src} = y\,e^{-k\,w(s)},\qquad s = x\,y,\quad w(s) = e^{-|s|/S} .$$

A naive global saddle ($w \equiv 1$) acts on the whole canvas; a naive radial window breaks incompressibility. Windowing by the streamline value $s = xy$ does neither: $s$ is conserved along every hyperbolic trajectory of the saddle, so the exponent is constant along any path and the map stays closed-form with $\det = 1$ exactly.

Thirty delta passes of the saddle: k per pass, fold axis θ on the top-left cluster and θ + 90° on the bottom-right; set v = 1 for the crossed-tines variant.open full size ↗

Invariants

• Exactly area-preserving — the Jacobian determinant is 1 everywhere, for any $k$.
• Honest caveat, stated as a feature: the window cannot decay along the two fold axes themselves ($s = 0$ there), so the pinch’s arms run outward as fading creases — which is what pinched paper physically does.
• $k$ is always a smoothed delta, never an absolute controller value: absolute feeding integrates into runaway strain. Its sign swaps which axis compresses — a naturally bipolar gesture.
• Two looks, one switch. pinch_variant 0 is this Hamiltonian saddle; 1 is crossed tines — two perpendicular, opposing tine passes through the pinch point. Softer, lumpier, directional; costs two passes per emission. It was the prototype rival kept after a pick-by-eye comparison, because the two are different instruments, not a better and a worse one.

The crossed-tines variant (v = 1), same k, θ and passes as the saddle above: two perpendicular opposing tine passes through each pinch point — softer, lumpier, directional, and not a saddle: the arms do not run to the edges.open full size ↗

Ownership

• CC 74 with slide_mode = 1: smoothed slide deltas drive the pinch at the drop (pitch-axis fold) — the ROLI’s vertical slide as a fold. With slide_mode = 0 (default) CC 74 keeps its per-drop hue role; the choice is a params setting, never a silent rebinding.
• The pen: with slide_mode = 1, the stylus’s CC 74 drives sumi_add_pinch at the pen position with the fold axis from the pen’s azimuth; that CC 74 then goes outbound only, because a looped copy would pinch twice.
• Two-finger pinch (tablets, Marble mode): the fold axis is the finger-to-finger line, $k$ comes from the scale deltas.
• Desktop: Shift-drag — drag distance is the strength delta, drag angle the fold axis.

In the scene

k is the per-pass strength (negative flips the axes), θ the fold angle, n how many delta passes are stacked, v the variant.