L-System Description of Subdivision Curves

Abstract

In recent years, subdivision has emerged as a major geometric modeling technique. Algorithms for generating subdivision curves are often specified in terms of iterated matrix multiplication. Each multiplication maps a globally indexed sequence of points that represents a coarser approximation of the curve onto a longer sequence that represents a finer approximation. Unfortunately, an infinite set of matrices is needed to specify these mappings for sequences of points of arbitrary length. Thus, matrix algebra is not well attuned to the dynamic nature of subdivision. In addition, matrix notation and the use of indices obscure the local and stationary character of typical subdivision rules.We introduce parametric context-sensitive L-systems with affine geometry interpretation as an alternative technique for specifying and generating subdivision curves. This technique is illustrated using Chaikin, cubic B-spline, and Dyn-LevinGregory (4-point) subdivision schemes as examples. L-systems formalize subdivision algorithms in an intuitive, concise, index-free manner, reflecting the parallel and local character of these algorithms. Furthermore, L-system specification of subdivision algorithms directly leads to their computer implementation.

Generating subdivision curves with L-systems on a GPU

“…Subdivision curves can be described using context-sensitive parametric L-systems [Prusinkiewicz et al 2003]. Control points of the curve are stored as symbols in the initial string, with parameters specifying point locations 1 .…”

Multilevel Editing of B-Spline Curves with Robust Orientation of Details

“…Prusinkiewicz et al [9] and Poon et al [10] presented a multiresolution representation in which a curve is subdivided by means of an L-system. Dreger et al [11] proposed a multiresolution representation of triangular B-spline surfaces of arbitrary degree.…”

Developmental Computing

“…Geometric examples include the generation of fractals [16] and space-filling curves [22], as well as the generation of smooth subdivision curves [23] and surfaces [26]. Similar to biological models, these algorithms involve a growing number of components and relations between them, and thus benefit from the conceptual and notational clarity afforded by developmental computing.…”