closed-form-marbling-renderer

name	closed-form-marbling-renderer
description	Render mathematically-faithful Turkish marbling (Ebru) and paper-marbling with closed-form coordinate-deformation maps — a drop is a radial outward push, a comb/tine is a decaying shear, all composed as homeomorphisms over colored region boundaries. No fluid solver, runs 60fps on a 2D canvas. Use when building a marbling/Ebru toy or generative-art renderer, implementing drop/comb/wave/swirl deformations, scripting traditional patterns (Battal, Gel-Git, Şal, Bülbül Yuvası), keeping combed edges crisp via adaptive resampling, modeling ox-gall shouldering for crisp ring-in-ring drops, or exporting high-DPI marbled prints. NOT for real fluid / Navier-Stokes simulation, metaball / alpha-blended "smoke" blob effects, raster image-processing filters, or non-marbling generative art.
allowed-tools	Read,Write,Edit,Bash
argument-hint	[pattern: battal|gelgit|sal|bulbul] [target: canvas2d|export]
metadata	{"category":"Creative & Visual","tags":["marbling","ebru","canvas","generative-art","closed-form-deformation"],"pairs-with":[{"skill":"physics-rendering-expert","reason":"Frame budget, transform math, and high-DPI canvas plumbing under the deformation algebra"},{"skill":"color-theory-palette-harmony-expert","reason":"Tuning period pigment hexes so they read as ground minerals, not electric CSS colors"}]}
category	Creative & Visual
tags	["marbling","ebru","closed-form","canvas","deformation"]

Closed-Form Marbling Renderer

Render real marbling the way Lu & Jaffer proved you can (IEEE CG&A 32(6), 2012): not by simulating fluid, but by treating the marbling tray as a set of colored region boundaries and applying each tool as a closed-form homeomorphism of the plane to every boundary point at once. A finished marble is just a composition Tn ∘ … ∘ T1 of these maps. Pure arithmetic, O(points) per op, 60fps.

When to use / when NOT

• Use to implement drop / tine / comb / wave / swirl ops, to script the four canonical Ebru patterns, to keep combed lines crisp, to model ox-gall, or to lift a high-res print off the tray.
• Do NOT reach for a fluid solver, reaction-diffusion, metaballs, or alpha-blended translucent disks. Those produce blurry "smoke goo," not the crisp veined cells of real Ebru, and they don't compose, replay, or scale to print.

Core mental model

flowchart LR
  S["State<br/>ops[] + regions[]"] --> D{Tool}
  D -->|drop| P["push ALL prior verts<br/>radially, then add<br/>fresh color circle"]
  D -->|tine / wave / swirl| W["warp ALL verts<br/>(closed form)"]
  W --> R["adaptive resample<br/>(keep edges crisp)"]
  P --> M["live regions"]
  R --> M
  M -->|painter's algo| C["Canvas2D (60fps)"]
  S -->|replay at print res| X["High-DPI PNG export"]

Four invariants that make this faithful instead of generic:

1. Boundaries, not pixels. Each color region is a closed polyline; tools move vertices. Smooth maps applied to dense polylines look like smooth dye.
2. Drops shoulder, they don't blend. A new drop physically pushes earlier ink aside (the radial map), so fills stay opaque and you get crisp ring-in-ring.
3. Everything composes. State is an ordered op list; you can replay, undo, serialize, and re-render at any resolution from the same script.
4. Resample under combing. Comb maps stretch segments a lot; subdivide or edges go faceted.

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The deformation algebra (copy-paste)

All functions take/return {x, y} and are pure. regions = [{ pts:[{x,y}…], color }], oldest first (paint order). Coordinates are in logical tray pixels.

1. Ink drop — radial outward push (area-preserving)

P' = C + (P − C) · sqrt( 1 + r² / ‖P − C‖² )

This is why nested drops form perfect concentric bullseyes: it maps circles to circles and is area-preserving away from . The only singularity is a vertex exactly at — guard with epsilon.