Simplicial homology

Topology

Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022

Steve Huntsman, Jimmy Palladino, Michael Robinson

The basic idea of simplicial homology is that structural deficits in abstract simplicial complexes can be straightforwardly and efficiently encoded in quotient vector spaces (or modules) indexed by the dimension of the deficit. In dimensions zero, one, and two, this means that connected components, holes, and ‘bubbles’ are identified by suitable representatives of the corresponding homology vector spaces, and the dimensions of these quotients give the numbers of the corresponding deficits.

A procedural footprint enhancement of global topographic surface with multiple levels of detail

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Journal Information

Published in International Journal of Digital Earth, 2020

Lukáš Brůha, Jan Kolář

Increased demand for 3D GIS applications stressed the structural complexity of pure 3D solutions. In order to avoid it, several solutions utilize the simpler 2D topology as much as possible. For example, an integrated TIN / tetrahedronized irregular network (TEN) model presented by Penninga (2005) only uses TEN representation for features such as bridges, which cannot be modeled by 2D TIN. Such 3D models are then put on top of the TIN using the footprint of the feature (e.g. Figure 2). The TEN concept was further elaborated in Penninga and Van Oosterom (2008) using the mathematical field of simplicial homology, which provides better control over orientation and boundary derivation of 3D features. These works, however, did not address issues related to multiple levels of detail.

Simplicial homology

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Topology

A procedural footprint enhancement of global topographic surface with multiple levels of detail