Semi-discrete principal net,
smoothed by channel surfaces
(Fig: M Lara Miro)
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Abstract.
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.
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