Recursive geometric definition of curves

The importance of forms as a tool for analysing and modelling nature has led to the study of shapes in many diverse fields of research and as a consequence many different mathematical description methods have been developed. This thesis focuses on the characterisation of forms using a recursive description, which provides a simple computational definition which can be used to model both regular and irregular forms.Accordingly, this thesis is divided into two main parts: firstly recursive shape definitions are used to create regular forms; and secondly irregular forms are modelled. In the modelling of regular forms, recursive characterisations can be defined by a subdivision process which represents a simple computational function of the form. In this thesis, the study of recursive regular curves focuses on the generation of models which correspond to generalised cylinders and superquadrics. Two novel rendering algorithms which exploit the geometry of the recursive definition of these structures are presented.In the study of irregular forms the definition of fractal geometry is considered. Irregular forms are characterised by the generalisation of geometry to recursive definitions. A novel description of irregular shapes by a functional equation is presented. This description is obtained by considering the frequency composition of shapes and is used to model and analyse static and dynamic irregular curves. Mappings between fractals are defined by considering the dynamic of irregular curves showing how a deformation of an irregular shape is possible. This represents an extension of fractal descriptions to animated forms.