In this paper we develop a new description for subdivision surfaces based on a graph grammar formalism. Subdivision schemes are specified by a context sensitive grammar in which production rules represent topological and geometrical transformations to the surface’s control mesh. This methodology can be used for all known subdivision surface schemes. Moreover, it gives an effective representation that allows simple implementation and is suitable for adaptive computations.

The introduction of floating−point pixel shaders has initiated a trend of moving algorithms from CPUs to graphics cards. The first algorithms were in the rendering domain, but recently we have witnessed increased interest in modeling algorithms as well. In this paper we present techniques for generating subdivision curves on a modern Graphics Processing Unit (GPU). We use an existing method for generating subdivision curves with L−systems, we extend these L−systems to implement adaptive subdivision, and we show how these L−systems can be implemented on a GPU. We chose L−systems because they can express many modeling algorithms in a compact way and are parallel in nature, making them an attractive paradigm for programming a GPU.

We present an important step towards the solution of the problem of inverse procedural modeling by generating parametric context-free L-systems that represent an input 2D model. The L-system rules efficiently code the regular structures and the parameters represent the properties of the structure transformations. The algorithm takes as input a 2D vector image that is composed of atomic elements, such as curves and poly-lines. Similar elements are recognized and assigned terminal symbols of an L-system alphabet. The terminal symbols ’ position and orientation are pair-wise compared and the transformations are stored as points in multiple 4D transformation spaces. By careful analysis of the clusters in the transformation spaces, we detect sequences of elements and code them as L-system rules. The coded elements are then removed from the clusters, the clusters are updated, and then the analysis attempts to code groups of elements in (hierarchies) the same way. The analysis ends with a single group of elements that is coded as an L-system axiom. We recognize and code branching sequences of linearly translated, scaled, and rotated elements and their hierarchies. The L-system not only represents the input image, but it can also be used for various editing operations. By changing the L-system parameters, the image can be randomized, symmetrized, and groups of elements and regular structures can be edited. By changing the terminal and non-terminal symbols, elements or groups of elements can be replaced.

Many polygon mesh algorithms operate in a local manner, yet are formally specified using global indexing schemes. This obscures the essence of these algorithms and makes their specification unnecessarily complex, especially if the mesh topology is modified dynamically. We address these problems by defining a set of local operations on polygon meshes represented by graph rotation systems. We also introduce the vv programming language, which makes it possible to express these operations in a machine−readable form. The usefulness of the vv language is illustrated by the application examples, in which we concentrate on subdivision algorithms for the geometric modeling of surfaces. The algorithms are specified as short, intuitive vv programs, directly executable by the corresponding modeling software.

In the areas of geometry and biology, there are a number of modelling problems that require the creation and manipulation of discrete surfaces that behave dynamically. For example, in geometric modelling there are surface subdivision algorithms that require the repeated insertion of vertices into a polygon mesh. In biological modelling there is the question of modelling growing surfaces, such as a growing flower or a growing tissue of cells. In these cases, there is the open question of how to model dynamical systems with a dynamical structure of a 2-manifold topology, discrete surfaces that have components that change in character, connectivity and number over time. However, the selection of available tools for modelling dynamical surfaces is lim-ited. There have been some proposed solutions for limited cases, such as cell systems for modelling cells. But there is still a need for a methodology and tools for dealing with dynamical surfaces in general. In this dissertation, I present a methodology for modelling dynamical systems

Figure 1: A top and perspective views of a fractal bicubic surface generated by an L-system Subdivision surfaces are becoming an important tool in Computer Graphics. They can be found in modern software as well as implemented in GPUs. However, their description is complex, and simple relations, as the inherently parallel manner of rewriting, are obfuscated by the indexing scheme. We propose using L-systems for tensor product surfaces subdivision description. The parallel rewriting helps us to merge the parallel manner of a surface subdivision. We demonstrate their functionality on Bézier bicubic and rational Bézier bicubic surfaces. We show that the power of L-systems can be easily applied to this scheme which is intuitive and easy to control. Parametric L-systems allow us to manage the level of subdivision and Open L-systems help to generate adaptive surfaces where the level of subdivision is controlled by an external condition. The ε-rule helps to generate surfaces with holes and in this way we can emboss classical fractals to 3D surfaces.

Abstract—In this paper, we propose a new method to describe fractal shapes using parametric l-systems. First we introduce scaling factors in the production rules of the parametric l-systems grammars. Then we decorticate these grammars with scaling factors using turtle algebra to show the mathematical relation between l-systems and iterated function systems (IFS). We demonstrate that with specific values of the scaling factors, we find the exact relationship established by Prusinkiewicz and Hammel between l-systems and IFS. Keywords—Fractal shapes, IFS, parametric l-systems, turtle algebra. I.

We show that parametric context-sensitive L-systems with affine geometry interpretation provide a succinct description of some of the most fundamental algorithms of geometric modeling of curves. Examples include the Lane-Riesenfeld algorithm for generating B-splines, the de Casteljau algorithm for generating Bézier curves, and their extensions to rational curves. Our results generalize the previously reported geometric-modeling applications of L-systems, which were limited to subdivision curves. Keywords: L-system; affine geometry; B-spline;Bézier curve; Lane-Riesenfield algorithm; de Casteljau algorithm; subdivision. 1.