Painting with Flowsnakes

Abstract

Space filling curves, invented by mathematicians in the 19th century, have long been a fascination for artists, however there are no interactive tools to allow an artist to create and explore various levels of recursion of the curve in different parts of the artwork. In this work a new type of painting tool for artists is introduced, which gives the artist control over the very base of a space filling curve, i.e recursive subdivision. Although there are many such curves that would lend themselves to this treatment, the Flowsnake (Gosper) curve has been chosen in this work, mainly for its aesthetics. The curve is based on a hexagonal grid, and in our system hexagons are subdivided at the artist's touch in a non-homogeneous manner, leaving a trail that forms the space filling curve. Some tools are introduced for controlling the painting, such as limiting the depth of recursion, and the 'slow brush', which interpolates slowly between subdivisions to allow the artist to stop at a chosen level. A set of space filling curve brush types provide different shapes and profiles, for giving the artist control of the nonhomogeneous subdivision, including the ability to un-subdivide the hexagons. An algorithm for drawing the curve non-recursively is introduced in order to produce a polyline suitable for processing on the GPU to make the system function at interactive rates. An animated version of the image can be made by replaying the subdivisions from the first level. Some examples made by art students and graduates are shown, along with the artist's comments on the system.