Designing 2D Ordinary Differential Equations To Obtain Abstract Paintings, Illustrations and Animations

Ergun Akleman and Hüseyin Koçak
Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015)
Pages 309–316


In this work, we introduce a simple method for designing ordinary differential equations that can provide desired motions in 2D. Our method provides a simple and intuitive way to construct desired vectors as a mixture of gradient and tangent fields. Both of these fields are defined using 2D implicit functions which can easily be built using existing methods for designing implicit curves. These differential equations can be used to obtain paintings from photographs, abstract illustrations and animations. For abstract animations, we can simply apply a designed differential equation to a set of particles. The motion of the particles directly provides an abstract 2D animation. The trajectories of particles can further be used as abstract illustrations. The particle motion can also be viewed as the motion of the hands of painters and the trajectories of the particles as long unbroken brush strokes over a photograph. These brush strokes can be used to turn photographs into paintings in a controlled way.