## L-SYSTEMS IN GEOMETRIC MODELING (2012)

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@MISC{Prusinkiewicz12l-systemsin,

author = {Przemyslaw Prusinkiewicz and Mitra Shirmohammadi},

title = { L-SYSTEMS IN GEOMETRIC MODELING},

year = {2012}

}

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### Abstract

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.

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Citation Context ...nientlyviewedasawordoveranalphabetof cellstates.Thisabstractionwasgeneralizedtobranchingstructuresandcomponents largerthan individualcells, leadingtoastonishinglysuccinct descriptionsofgrowing plants =-=[27]-=-. 133134 P. Prusinkiewicz, M. Shirmohammadi & F. Samavati In retrospect, the success of L-systems in plant modeling can be attributed to the following factors [24]: (1) L-systemsspecifydevelopmentint... |

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Citation Context ...enfield algorithm; de Casteljau algorithm; subdivision. 1. Introduction L-systems were conceived by Aristid Lindenmayer as a mathematical formalism for reasoning about growing multicellular organisms =-=[17]-=-. They were originally introduced as an extension of cellular automata, allowing for the addition and removal of cells during an automaton’s operation, but were soon rephrased in terms of rewriting sy... |

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Citation Context ...escription of linear fractals, including classic space-filling curves [22,28,34,36]. The geometric interpretations of L-systems used in these studies were based on turtle geometry [1] or chain coding =-=[9]-=-. More recently, L-systems with an interpretation based on affine geometry [6,7,11,12] have been demonstrated to provide a concise description of the subdivision curves used in geometric modeling [23,... |

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Citation Context ...otation borrowed from formal language theory. The relation to formal languages colored early studies of L-systems, which were often focused on their ability to generate different classes of languages =-=[13,32]-=-. Concurrent with these developments, although initially on a smaller scale, Lsystems began to be used as a mathematical foundation for the computational modeling ofplants. The fundamental observation... |

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Citation Context ...step in L-system (7) (compare the middle row in Figure 3 with Figure 2). Lane and Riesenfeld have shown that the polygons generated by L-system (9) converge to (uniform) B-spline curves of degree n+1 =-=[16]-=- (see also [20,35]). B-splines are widely used in geometric modeling due to their well understood geometric properties, and the relative ease with which diverse curves can be defined by interactively ... |

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Citation Context ...(v), divides line P(v1)P(v2) in proportion α2 : α1 (Figure 1). 3. B-Splines A simple example of the application of L-systems to geometric modeling is an Lsystem specification of the Chaikin algorithm =-=[2]-=-. Given a (closed) control polygon P(v0)P(v1)...P(vm), this algorithm produces a smooth (at the limit) curve, whose shape can be thought of as the result of iteratively cutting the cornersof the contr... |

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Citation Context ...e concise L-system description of linear fractals, including classic space-filling curves [22,28,34,36]. The geometric interpretations of L-systems used in these studies were based on turtle geometry =-=[1]-=- or chain coding [9]. More recently, L-systems with an interpretation based on affine geometry [6,7,11,12] have been demonstrated to provide a concise description of the subdivision curves used in geo... |

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Citation Context ... rules to discrete structures with a changing number of components. Early studies of such algorithms led to the concise L-system description of linear fractals, including classic space-filling curves =-=[22,28,34,36]-=-. The geometric interpretations of L-systems used in these studies were based on turtle geometry [1] or chain coding [9]. More recently, L-systems with an interpretation based on affine geometry [6,7,... |

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Citation Context ...otation borrowed from formal language theory. The relation to formal languages colored early studies of L-systems, which were often focused on their ability to generate different classes of languages =-=[13,32]-=-. Concurrent with these developments, although initially on a smaller scale, Lsystems began to be used as a mathematical foundation for the computational modeling ofplants. The fundamental observation... |

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Citation Context ... were originally introduced as an extension of cellular automata, allowing for the addition and removal of cells during an automaton’s operation, but were soon rephrased in terms of rewriting systems =-=[18]-=-. This rephrasing led to an elegant definition of L-systems using notions and notation borrowed from formal language theory. The relation to formal languages colored early studies of L-systems, which ... |

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Citation Context ... by Shirmohammadi [33], whose workwas a stepping stone for the present paper. Compared to [33], we use a topologically more accurate representation of polygons, rooted in the notion of a cell complex =-=[21]-=-, with both vertices and edges explicitly represented. This results in a more intuitive specifications of the algorithms, reflecting their inherent symmetries.L-Systems in Geometric Modeling 135 2. P... |

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Citation Context ...v) carries all the information needed to visualize it as, for example, a smallcirclecenteredatv. In contrast,the visualizationofthe edgeE betweenpoints P(vl) and P(vr) requires an interpretation rule =-=[14,15]-=- that is applied at the end of the derivation and gathers the information about the line’s endpoints: hE : P(vl) < E > P(vr) → L(vl,vr). (8) Module L now carries all the information needed to draw a l... |

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Citation Context ...m (7) (compare the middle row in Figure 3 with Figure 2). Lane and Riesenfeld have shown that the polygons generated by L-system (9) converge to (uniform) B-spline curves of degree n+1 [16] (see also =-=[20,35]-=-). B-splines are widely used in geometric modeling due to their well understood geometric properties, and the relative ease with which diverse curves can be defined by interactively positioning the co... |

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Citation Context ...ive specifications of the algorithms, reflecting their inherent symmetries.L-Systems in Geometric Modeling 135 2. Preliminaries The L-systems employed here are context-sensitive parametric L-systems =-=[19,27]-=-, extended in two directions: • Parameters are not limited to numbers, but may also be compound data structures, for example representing points or vectors in two or three dimensions [14,29]; • Produc... |

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Citation Context ...e representing points or vectors in two or three dimensions [14,29]; • Productionsmaybegroupedintosubsets(tables[31]),whereacontrolmechanism decides which subset is applicable in each derivation step =-=[3,4,5,10,26,37]-=-. The key concepts can be summarized as follows. Parametric L-systems operate on parametric words, or strings of modules. Each module is a letter from the Lsystem alphabet V, which may be associated w... |

19 | A unified subdivision scheme for polygonal modeling
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Citation Context ...m (7) (compare the middle row in Figure 3 with Figure 2). Lane and Riesenfeld have shown that the polygons generated by L-system (9) converge to (uniform) B-spline curves of degree n+1 [16] (see also =-=[20,35]-=-). B-splines are widely used in geometric modeling due to their well understood geometric properties, and the relative ease with which diverse curves can be defined by interactively positioning the co... |

18 |
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Citation Context ...er needed. These features are not only important to the modeling of biological development, but also make L-systems well suited to the description and implementation of some non-biological algorithms =-=[25]-=-. Typically, they are characterized by the repetitive application of relatively simple rules to discrete structures with a changing number of components. Early studies of such algorithms led to the co... |

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Citation Context ... [9]. More recently, L-systems with an interpretation based on affine geometry [6,7,11,12] have been demonstrated to provide a concise description of the subdivision curves used in geometric modeling =-=[23,29]-=-. Similar to the algorithms for generating fractal curves, subdivision algorithms operate on discrete, polygonal lines with a changing number and configuration of components. The rules of change can b... |

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Citation Context ...4,36]. The geometric interpretations of L-systems used in these studies were based on turtle geometry [1] or chain coding [9]. More recently, L-systems with an interpretation based on affine geometry =-=[6,7,11,12]-=- have been demonstrated to provide a concise description of the subdivision curves used in geometric modeling [23,29]. Similar to the algorithms for generating fractal curves, subdivision algorithms o... |

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Citation Context ...e representing points or vectors in two or three dimensions [14,29]; • Productionsmaybegroupedintosubsets(tables[31]),whereacontrolmechanism decides which subset is applicable in each derivation step =-=[3,4,5,10,26,37]-=-. The key concepts can be summarized as follows. Parametric L-systems operate on parametric words, or strings of modules. Each module is a letter from the Lsystem alphabet V, which may be associated w... |

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Citation Context ... rules to discrete structures with a changing number of components. Early studies of such algorithms led to the concise L-system description of linear fractals, including classic space-filling curves =-=[22,28,34,36]-=-. The geometric interpretations of L-systems used in these studies were based on turtle geometry [1] or chain coding [9]. More recently, L-systems with an interpretation based on affine geometry [6,7,... |

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Citation Context ...4,36]. The geometric interpretations of L-systems used in these studies were based on turtle geometry [1] or chain coding [9]. More recently, L-systems with an interpretation based on affine geometry =-=[6,7,11,12]-=- have been demonstrated to provide a concise description of the subdivision curves used in geometric modeling [23,29]. Similar to the algorithms for generating fractal curves, subdivision algorithms o... |

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Citation Context ...4,36]. The geometric interpretations of L-systems used in these studies were based on turtle geometry [1] or chain coding [9]. More recently, L-systems with an interpretation based on affine geometry =-=[6,7,11,12]-=- have been demonstrated to provide a concise description of the subdivision curves used in geometric modeling [23,29]. Similar to the algorithms for generating fractal curves, subdivision algorithms o... |

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Citation Context ...erate equally well in three and more dimensions. This provides a means of generating curves in space, and also leads to a useful extension of Bézier and B-spline curves to their rational counterparts =-=[8,30]-=-. Rational curves are generated in a higher-dimensional space, then projected to lower dimensions using a perspective projection. Assuming z ̸= 0, the projection of a 3D point P(x,y,z) on the plane z ... |

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Citation Context ... rules to discrete structures with a changing number of components. Early studies of such algorithms led to the concise L-system description of linear fractals, including classic space-filling curves =-=[22,28,34,36]-=-. The geometric interpretations of L-systems used in these studies were based on turtle geometry [1] or chain coding [9]. More recently, L-systems with an interpretation based on affine geometry [6,7,... |

3 | Developmental computing
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Citation Context ...ccinct descriptionsofgrowing plants [27]. 133134 P. Prusinkiewicz, M. Shirmohammadi & F. Samavati In retrospect, the success of L-systems in plant modeling can be attributed to the following factors =-=[24]-=-: (1) L-systemsspecifydevelopmentintermsoftemporallyandspatiallylocaldeclarative rewriting rules — context-free and context-sensitive productions — which express developmental processes in a concise a... |

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Citation Context ...e representing points or vectors in two or three dimensions [14,29]; • Productionsmaybegroupedintosubsets(tables[31]),whereacontrolmechanism decides which subset is applicable in each derivation step =-=[3,4,5,10,26,37]-=-. The key concepts can be summarized as follows. Parametric L-systems operate on parametric words, or strings of modules. Each module is a letter from the Lsystem alphabet V, which may be associated w... |

2 |
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Citation Context ...y concise description of this class of curves. 4. The de Casteljau Algorithm The de Casteljau algorithm is considered “probably the most important algorithm of all of computer-aided geometric design” =-=[8]-=- . Given an open control polygon P(v0 1 )P(v0 2 )...P(v0 n ) with n ≥ 2 vertices P located at v0 1 ,v0 2 ,...,v0 n , the algorithm ) subdivides constructs a polygon P(v1 1 )P(v1 2 )...P(v1 n−1 ) such ... |

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Citation Context ... [9]. More recently, L-systems with an interpretation based on affine geometry [6,7,11,12] have been demonstrated to provide a concise description of the subdivision curves used in geometric modeling =-=[23,29]-=-. Similar to the algorithms for generating fractal curves, subdivision algorithms operate on discrete, polygonal lines with a changing number and configuration of components. The rules of change can b... |

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Citation Context ...senfeld algorithm for generating B-splines, the de Casteljau algorithm for generating Bézier curves, and their extensions to rational curves. Further examples are discussed in detail by Shirmohammadi =-=[33]-=-, whose workwas a stepping stone for the present paper. Compared to [33], we use a topologically more accurate representation of polygons, rooted in the notion of a cell complex [21], with both vertic... |

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