Shape computations are a formal representation that specify particular aspects of the design process with reference to form. They are defined according to shape grammars, where manipulations of pictorial representations of designs are formalised by shapes and rules applied to those shapes. They have frequently been applied in architecture in order to formalise the stylistic properties of a given corpus of designs, and also to generate new designs within those styles. However, applications in more general design fields have been limited. This is largely due to the initial definitions of the shape grammar formalism which are restricted to rectilinear shapes composed of lines, planes or solids. In architecture such shapes are common but in many design fields, for example industrial design, shapes of a more freeform nature are prevalent. Accordingly, the research described in this thesis is concerned with extending the applicability of the shape grammar formalism such that it enables computation with freeform shapes.

Shape computations utilise rules in order to manipulate subshapes of a design within formal algebras. These algebras are specified according to embedding properties and have previously been defined for rectilinear shapes. In this thesis the embedding properties of freeform shapes are explored and the algebras are extended in order to formalise computations with such shapes. Based on these algebras, shape operations are specified and algorithms are introduced that enable the application of rules to shapes composed of freeform B´ezier curves. Implementation of the algorithms enables the application of shape grammars to shapes of a more freeform nature than was previously possible. Within this thesis shape grammar implementations are introduced in order to explore both theoretical issues that arise when considering computation with freeform shapes and practical issues concerning the application of shape computation as a model for design and as a mode for generating freeform shapes.

Computation with Curved Shapes: Towards Freeform Shape Generation in Design

This paper proposes a parametric solid representation of freeform tubular objects - Ball B-Spline Curves (BBSCs), which are skeleton based parametric solid model. Their fundamental properties, algorithms and modeling methods are investigated. BBSC directly defines objects in B-Spline function form (not a procedure method, like sweeping), that uses control ball instead of control point in B-Spline curve. BBSC describes not only every point inside 3D solid objects, but also provides its center curve in B-Spline form directly. So the representation is more flexible for modeling, manipulation and deformation.

Skeleton Based Parametric Solid Models: Ball B-Spline Surfaces

In this paper we consider error propagation in geometric constructions from a geometric viewpoint. First we study affine combinations of convex bodies: this has many applications concerning spline curves and surfaces defined by control points. Second, we study in detail the circumcircle of three points in the Euclidean plane. It turns out that the right geometric setting for this problem is Laguerre geometry and the cyclographic mapping, which provides a point model for sets of circles or spheres.

Error propagation in geometric constructions

Disk B-spline curve (DBSC) is previously proposed for drawing and animation. To generate inbetweens, linear interpolation is applied between points evenly taken in parametric domain of two DBSCs without incorporating characteristics of shape or motion, which results in distortion and unrealistic motion in animation. In this paper, more information in keyframes is extracted and utilized in inbetween generation. Points with high curvature are computed and corresponded between strokes in interpolation, which preserves features of strokes in animation. In addition, global motion of a character or its various components is estimated and interpolated as well, which retains the shapes during the motion. By applying the information to interpolation, the distortion is eliminated and smoother sequence of animation is achieved. Copyright © 2006 John Wiley & Sons, Ltd.

DBSC-based animation enhanced with feature and motion

Degree reduction of disk Bézier curves

Disk Be´zier curves

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Disk Be´zier curves