“Mathematical Marbling”, 2011-11 ():
In this paper, the proposed method takes a mathematical approach with closed-form expressions to simulate marbling. This method improves control, ease of implementation, parallelism, and speed, enabling real-time visual feedback and creation of vivid flowing animations. Users can start designs from a blank sheet, raster images, or videos.
…Marbling creates stone-like or intricate abstract decorations from inks (or paint) floating on water or gel. It’s a decorative art with several distinct traditions originating in Asia, perhaps as many as 1,000 years ago. It spread to Europe in the 16th century, where its primary application was producing book covers and endpapers. Mechanized bookbinding caused the decline of marbling in the West, but it has enjoyed a revival as a folk art since the 1970s. Although primarily used for decoration, marbling has security applications. Marbling ledger book edges makes missing pages apparent, and documents written over pale marbling are tamper-resistant.
Digital simulations based on complex physical models have been commonly used to create marbling images.1,2 However, these methods produce blurry contours because the time-iterative-relaxation nature of the solver makes dissipation inevitable. The more marbling operations are applied, the blurrier the result is. So, these methods have difficulty producing publication-quality images because fine features will be lost. (For more on digital marbling methods, see the related sidebar.) This motivates us to find simple closed-form mathematical formulas to simulate marbling.
Here, we present deformation formulas for simulating marbling, while avoiding the computational cost of full fluid simulation. This approximation is rich enough to capture many phenomena, and it solves the dissipation problem and ensures the resulting images’ sharp contours. Besides simplicity, using mathematical formulas provides advantages for control, speed, implementation ease, parallelism, and vector output. It enables the generation of beautiful designs with real-time visual feedback and progressive fluid-like illustration of marbling.
Our Method: Our mathematical treatment of marbling starts with the assumptions of incompressible and immiscible 2D fluid inks. Our tool function formulas are based on topological computer graphics, 3 which generates marbling designs with sharp contours and vector marbling output.
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