Tool-path generation for fractal curve making

Many fractal generation methods have been developed and used to create an image of a natural scene. Nonlinear dynamic systems employ fractal theory for population growth. Fractals have also been used to model chaotic problems. In numerical control (NC) machining, fractal curves have been used in tool-path generation. Although the visualisation of fractal geometry has been successfully demonstrated by computer graphics, a manufacturing method for physical fractal objects is not available. Moreover, contemporary computer-aided design (CAD) systems consider only Euclidean geometry and none of them addresses fractal geometry. Fractal curves have been used in tool-path planning for Euclidean objects, but there is no report on rapid prototyping (RP) of objects defined in fractal geometry. In the paper, a new data structure, called the radial-annular tree (RAT) structure, is proposed and implemented to bridge the gaps between CAD, RP, and fractal geometry. A typical fractal curve, the Koch snowflake curve, will be examined in detail. Based on the RAT representation, higher-level fractal curves can be generated more efficiently, and repeated information can be represented concisely. Traversal algorithms are also devised to generate a maximally connected tool path directly. The tool path can then be used to generate a physical fractal curve without any additional conversion.