Geometry of discrete and continuous bounded surfaces

We work on reconstructing discrete and continuous surfaces with boundaries using length constraints. First, for a bounded discrete surface, we discuss the rigidity and number of embeddings in three-dimensional space, modulo rigid transformations, for given real edge lengths. Our work mainly considers the maximal number of embeddings of rigid graphs in three-dimensional space for specific geometries (annulus, strip). We modify a commonly used semi-algebraic, geometrical formulation using Bézout\u27s theorem, from Euclidean distances corresponding to edge lengths. We suggest a simple way to construct a rigid graph having a finite upper bound. We also implement a generalization of counting embeddings for graphs by segmenting multiple rigid graphs in d-dimensional space. Our computational methodology uses vector and matrix operations and can work best with a relatively small number of points

Rethinking the Expressive Power of GNNs via Graph Biconnectivity

Designing expressive Graph Neural Networks (GNNs) is a central topic in learning graph-structured data. While numerous approaches have been proposed to improve GNNs in terms of the Weisfeiler-Lehman (WL) test, generally there is still a lack of deep understanding of what additional power they can systematically and provably gain. In this paper, we take a fundamentally different perspective to study the expressive power of GNNs beyond the WL test. Specifically, we introduce a novel class of expressivity metrics via graph biconnectivity and highlight their importance in both theory and practice. As biconnectivity can be easily calculated using simple algorithms that have linear computational costs, it is natural to expect that popular GNNs can learn it easily as well. However, after a thorough review of prior GNN architectures, we surprisingly find that most of them are not expressive for any of these metrics. The only exception is the ESAN framework (Bevilacqua et al., 2022), for which we give a theoretical justification of its power. We proceed to introduce a principled and more efficient approach, called the Generalized Distance Weisfeiler-Lehman (GD-WL), which is provably expressive for all biconnectivity metrics. Practically, we show GD-WL can be implemented by a Transformer-like architecture that preserves expressiveness and enjoys full parallelizability. A set of experiments on both synthetic and real datasets demonstrates that our approach can consistently outperform prior GNN architectures.Comment: ICLR 2023 notable top-5%; 58 pages, 11 figure

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Mathematical surfaces models between art and reality

In this paper, I want to document the history of the mathematical surfaces models used for the didactics of pure and applied “High Mathematics” and as art pieces. These models were built between the second half of nineteenth century and the 1930s. I want here also to underline several important links that put in correspondence conception and construction of models with scholars, cultural institutes, specific views of research and didactical studies in mathematical sciences and with the world of the figurative arts furthermore. At the same time the singular beauty of form and colour which the models possessed, aroused the admiration of those entirely ignorant of their mathematical attraction

Blending techniques in Curve and Surface constructions

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Play Among Books

How does coding change the way we think about architecture? Miro Roman and his AI Alice_ch3n81 develop a playful scenario in which they propose coding as the new literacy of information. They convey knowledge in the form of a project model that links the fields of architecture and information through two interwoven narrative strands in an “infinite flow” of real books

Play Among Books

How does coding change the way we think about architecture? Miro Roman and his AI Alice_ch3n81 develop a playful scenario in which they propose coding as the new literacy of information. They convey knowledge in the form of a project model that links the fields of architecture and information through two interwoven narrative strands in an “infinite flow” of real books