Geometric shape generation

Explicit classification results and representation formulae are at the core of the differential geometry of curves and surfaces - they serve to generate geometric shapes (curves or surfaces) with certain prescribed properties: for example, the classical Weierstrass representation formulae serve to generate any surface that (locally) minimizes area out of simple data. Other shape generation methods include "transformations", which transform a given shape of a certain class into another such shape, while preserving its key properties. While such "shape generation methods" are designed to produce curves or surfaces of a particular kind out of suitable input data, it is often difficult to control other features of the generated shape by the input data - deep knowledge about the particular shapes and the generation process are required. These shape generation methods play an important role in geometry, not just for the production of interesting shapes for design or ilustration purposes, but also to obtain a better understanding of the structure of the investigated shapes. In particular, the properties of transformations are essential for describing facetted or panelled surfaces that display similar properties as the corresponding smooth surfaces. In this project we aim to investigate different methods to generate shapes, in particular: the interrelations between different shape generation methods; the related discretizations and, hence, discretizations of the shape generation methods; the applicability and scope of these shape generation methods in theory and generative art and design. By interlinking these different aspects of shape generation we hope and expect to gain new insight and to establish new interesting methods for the geometric generation of shapes, for their use in theory as well as for their application in art or design. The planned collaboration between researchers in Japan and in Austria will be key to the success of the project: we will be able to rely on a unique combination of expertise, from various areas relating to the main questions of the project. Our strategy to focus funding on longer research visits of young researchers will help to create sustainable and long term research collaborations.

Explicit classifications and representation formulae are at the core of the differential geometry of curves and surfaces - they serve to generate geometric shapes (curves or surfaces) with certain prescribed properties: for example, any surface that (locally) minimizes area can be constructed out of simple input data, using the classical Weierstrass representation formulae. Other shape generation methods include "transformations", which transform a given shape of a certain class into another such shape, while preserving its key properties. While such "shape generation methods" are designed to produce curves or surfaces of a particular kind out of suitable input data, it is often difficult to control other features of the generated shape by the input data - deep knowledge about the particular shapes and the generation process are required. These shape generation methods play an important role in geometry, not just for the production of interesting shapes for design or illustration purposes, but also to obtain a better understanding of the structure of the investigated shapes. In particular, the properties of transformations are essential for describing facetted or panelled surfaces that display similar properties as the corresponding smooth surfaces. In this project investigated different methods to generate shapes, with a particular view on: - the interrelations between different shape generation methods; - the related facetted or panelled shapes and, hence, discrete versions of the investigated shape generation methods; - the applicability and scope of these shape generation methods in theory and generative art and design. More specifically, - we were able to find new methods to produce shapes for which no simple shape generation process was available before, in particular also for various classes of (discrete) panelled surfaces; - the relations between different previously known shape generation methods provided beautiful links and a deep insight into the geometry of the generated shapes in hyperbolic geometry; - we obtained deeper insight into the generation of circular and cyclidic nets, that may be useful for design in applications, and a prototype software library to facilitate such design processes has been implemented; - the investigated shape generation methods have been applied to get a better understanding of various open problems in geometry. On the other hand, the project has also opened up new ideas for research questions and strategies to solve old problems. The close collaboration between researchers in Japan and Austria has created a unique combination of expertise in various areas, that was key to the success of the project. The collaboration has also facilitated the creation of new research questions and projects, especially for young project collaborators, hence has contributed to create a sustainable research environment.