# Postdoc proposal

Topological properties and shape characteristics for idealized models

## Advisors

Jean-Claude Léon | Franck Hétroy |

SIREP/G-SCOP | LJK/EVASION |

E-mail : Jean-Claude.Leon@grenoble-inp.fr | E-mail : Franck.Hetroy@imag.fr |

Tel. : 04 76 82 51 27 | Tel. : 04 76 61 55 04 |

## Context

Shape characterization for non-manifold models is based on geometrical and topological concepts. In the past, topological aspects have been adressed from both a local point of view, with data structures conveying the connections between the components of non-manifold models, and a global one, with the Euler-Poincaré theorem and Betti numbers. The main difficulty with respect to non-manifold models is to get global topological characteristics which are related to their shape. Such topological parameters help classifying objects, thus distinguishing them.

## Objectives

The main stages of this postdoctoral project are the following:

- participation to the study [LF08, LFH09] of shape classification for manifold-connected components [HF07] of
a non-manifold simplicial complex:
- identification of topological properties of non-manifolds derived from 2-manifold with singularities;
- study of different classes of non-manifold objects, with respect to the orientation notion;
- specification of a generic data structure to describe non-manifold objects, based on these properties;
- creation of algorithms to compute shape characteristics for manifold-connected components;

- participation to the study, initiated by J.C. Léon and L. de Floriani, of properties of non-manifold
objects made of several manifold-connected components, based on hypergraphs:
- adaptation to objects with several manifold-connected components of global topological properties identified for objects with a single manifold-connected component;
- specification a data structure to describe non-manifold objects, based on these properties;
- creation of algorithms to compute shape characteristics for non-manifold objects made of several manifold-connected components.

## Pre-requisites

Candidates must have defended a PhD thesis in computer science or applied maths, and have a strong background on computational geometry.
Required skills also include geometric modeling, computer graphics, object-oriented programming and the ability to work in a team in an international
context. Candidates must agree to spend several weeks in Genova to work with our Italian partner.

**Keywords** : topology, non-manifold, singularity, manifold-connected component, Betti number, hypergraph, knot theory.

## Application conditions, duration and salary

To apply, candidates must:- be under 35;
- have defended their PhD thesis less than 3 years ago;
- have carried out research work outside Grenoble.

## References

- [HF07] A. Hui and L. de Floriani. A two-level topological decomposition for non-manifold simplicial shapes. Solid and Physical Modeling Conference, 2007.
- [LF08] J.C. Léon and L. de Floriani. Contribution to a taxonomy of non-manifold models based on topological properties. ASME DETC CIE conference, 2008.
- [LFH09] J.C. Léon, L. de Floriani and F. Hétroy. Classification of non-manifold singularities from transformations of 2-manifolds. IEEE International Conference on Shape Modeling and Applications (SMI), 2009.