Exploiting lattice structures in shape grammar implementations

Abstract

The ability to work with ambiguity and compute new designs based on both defined and emergent shapes are unique advantages of shape grammars. Realizing these benefits in design practice requires the implementation of general purpose shape grammar interpreters that support: (a) the detection of arbitrary subshapes in arbitrary shapes and (b) the application of shape rules that use these subshapes to create new shapes. The complexity of currently available interpreters results from their combination of shape computation (for subshape detection and the application of rules) with computational geometry (for the geometric operations need to generate new shapes). This paper proposes a shape grammar implementation method for three-dimensional circular arcs represented as rational quadratic Bézier curves based on lattice theory that reduces this complexity by separating steps in a shape computation process from the geometrical operations associated with specific grammars and shapes. The method is demonstrated through application to two well-known shape grammars: Stiny's triangles grammar and Jowers and Earl's trefoil grammar. A prototype computer implementation of an interpreter kernel has been built and its application to both grammars is presented. The use of Bézier curves in three dimensions opens the possibility to extend shape grammar implementations to cover the wider range of applications that are needed before practical implementations for use in real life product design and development processes become feasible.

Research on National Pattern Reuse Design and Optimization Method Based on Improved Shape Grammar

“…The Figure 7(a) pattern element is set as pattern 1 , and the Figure 7(b) pattern element is set as pattern 2 . First, the predicate shape grammar vocabulary V pattern 1 and V pattern 2 are defined, as shown in Equations (8) and (9), and then the pattern elements of Figure 7(a) and (b) are transformed according to the rules of V pattern 1 and V pattern 2 . The result is shown in Figure 8.…”

“…In the literature, research on combining shape grammar with other optimization algorithms focuses on combining shape grammar with parallel particle swarm optimization (PSO), which was proposed by Keles [7], in order to solve the problem of embedding and replacing the emergent part after the design is generated with shape grammar. Chau et al [8] proposed a method that combines shape grammar with lattice theory and uses lattice-based rational quadratic Bézier curves to represent three-dimensional circular arc shape grammar, which solves the complexity of geometric operations related to specific syntactic shapes in shape computation. Jia and Ming [9] proposed the dynamic shape representation based on shape grammar and extent the theory of shape grammar and further enriches the form of grammar rules.…”

A lattice-based approach for navigating design configuration spaces

“…The realisation of shape grammar-based design synthesis requires a means of implementing the embedding of shapes into each other; Krstic and Chau et al outline hypercube lattice-based solutions to this problem [17,18]. March [19] used lattices to describe geometric shapes, Stiny [16] used a lattice of parts to describe continuity in a sequence of shape rule applications and Krstic [17] used lattices in shape decompositions.…”

Necessary and Sufficient Conditions for Expressing Quadratic Rational Bézier Curves

“…Quadratic rational Bézier curves can be used to represent an arc if and only if |P 0 P 1 | = |P 2 P 1 | and 0 ≤ ω ≤ 1[25].…”