The Construction of Curved Shapes

Abstract

Application of a shape grammar involves the repetitive task of matching and replacing subshapes of a design under transformation, and as such is well suited for computer implementation. As a result, ever since the conception of the shape grammar formalism, efforts have been made to develop computer programs that automate shape grammar applications. Much of this effort has been directed towards the problem of subshape detection, which involves recognising subshapes embedded in a design. Solutions to this problem have been presented for shapes composed of rectilinear geometric elements, such as straight lines, and algorithms based on these solutions have been implemented in a variety of shape grammar interpreters. However, there has been less research concerning the solution of the subshape detection problem for shapes composed of nonrectilinear geometric elements, such as curve segments. In this paper a method of intrinsic matching is presented, which enables comparison of the embedding properties of parametric curves. This method has been employed in order to develop shape algorithms which can be implemented in shape grammar interpreters for shapes composed of parametric curve segments, arranged in two-dimensional or three-dimensional space.

Exploiting lattice structures in shape grammar implementations

“…Two examples, the triangles grammar (Stiny, 1994) and the trefoil grammar (Jowers & Earl, 2010), are used to show how steps 2 and 3 are performed in practice.…”

“…The first three implementations (Krishnamurti, 1981; Chase, 1989; Tapia, 1999) use two endpoints to describe straight lines in two dimensions ( U 12 ). The other two (Li et al, 2009; Jowers & Earl, 2010) use circular arcs and curves, respectively, with the latter using parametric Bézier curves or their variations to describe curves.…”

“…This paper provides a mechanism where lattices are used to reduce this complexity by decoupling computational geometry (needed for the geometrical operations associated with specific grammars used in the generation of new shapes) and shape computation (needed for sub-shape detection and the application of rules). A prototype implementation of an interpreter kernel has been built and applied to two well-known shape grammars: Stiny's triangles grammar (Stiny, 1994) and Jowers and Earl's trefoil grammar (Jowers & Earl, 2010). Early results are promising with more applications, for wider evaluation, and integration with suitable user interfaces as important next steps.…”

Extending the Algebras of Design

“…With respect to implementation, it is possible to utilise either analytical or visual embedding. For example, in (Jowers and Earl 2010), a method of implementing analytical embedding is presented, which builds on the intrinsic properties of parametric curves. Conversely, in ), a method of implementing visual embedding is presented, which builds on techniques of computer vision for object recognition in bitmap images.…”

“…Instead, other descriptors are necessary for distinguishing between types of geometric elements according to embedding properties. For example, in (Jowers and Earl 2010) intrinsic properties are used to distinguish between geometric elements of different types.…”