Research project funded by the Austrian science fund FWF.

Project details

FWF project: P 29981Livetime: 2017/09/01 - 2023/2/28

www: http://www.geometrie.tuwien.ac.at/ig/mueller/projects/pmc/

People

Project leader: | Christian Müller |

Project members: | Christian Müller |

Klara Mundilova | |

Michael Jimenez | |

Felix Dellinger | |

Efilena Baseta | |

Kurt Leimer |

Abstract

Discrete surfaces have been investigated since thousands of years. The
interest in those surfaces is theoretical, philosophical and also with a
view towards applications. In particular in architecture and civil
engineering appear discrete surfaces which can be seen as surfaces
composed of individual facets. These facets can be planar triangles,
quadrilaterals or other polygons. In some cases it makes even sense to
consider non-planar facets.

A surface in three dimensional space, discrete or otherwise, has some
curvature. There are several notions of curvature. In our project we are
interested in the so called mean curvature. For example, a plane has mean
curvature 0 whereas a sphere of radius r has mean curvature 1/r. A large
radius implies a small curvature.

In the last decades discrete analogs of smooth curvature notions have
been investigated. There are several discrete mean curvature notions
which illustrates that the process of discretization is not always
unique. The branch of mathematics that studies these objects and in which
the present proposal is located is called “discrete differential
geometry”.

It now seems natural to analyze a given discrete surface and to determine
its curvature. Our primary objective is the reverse question. Suppose we
are given any arbitrary mean curvature function. Is there a surface which
has exactly that curvature? In smooth differential geometry this type of
questions is by now very well studied even though there are still open
questions. In the setting of discrete surfaces this question has been
addressed only for two special cases which are geometrically important,
though.

The question is, how does a surface, smooth or discrete, look like if the
curvature is zero or any other constant everywhere. Both cases appear in
nature. For example, as soap films in equilibrium. If the air pressure
is equal on both sides of the surface then we have vanishing mean
curvature and otherwise the mean curvature is any other constant number.

In our project we prescribe a mean curvature function which is not
constant. A typical question can be the following. Suppose we are given a
function which maps a value to every point in three dimensional space.
This function will be our prescribed mean curvature function. Further we
choose a curve in space which can be thought as a wire loop. Now
investigate the existence of a surface which passes at the end through
that curve such that in between the mean curvature value of that surface
is exactly equal to the value of our prescribed function.

Publications:

*SIAM J. Discrete Math.*35-2 (2021), pp. 637-660.

*Ann. Mat. Pura Appl.*, accepted (2021).

*ArXiv e-prints*, 2009.05469 (2020).

*Advances in Architectural Geometry 2020*.

*ACM Trans. Graphics*

**39**, 4 (2020), 127:1-127:17, Proc. SIGGRAPH.

*Discrete Comput. Geom.*

**63**(3), (2020), 670-704.

*ACM Trans. Graphics*

**38**,6 (2019), 173:1-173:13, Proc. SIGGRAPH ASIA.

*SIAM J. Discrete Math.*35-2 (2021), pp. 637-660.

*ACM Trans. Graphics*(Proc. SIGGRAPH ASIA),

**37**(6), 227:1-227:15, 2018.

*Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture*, (2018), 535-538.