Abstract
Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the Ω-dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorphic maps.
| Original language | English |
|---|---|
| Pages (from-to) | 816-844 |
| Number of pages | 29 |
| Journal | Discrete and Computational Geometry |
| Volume | 70 |
| Issue number | 3 |
| Early online date | 20 Oct 2022 |
| DOIs | |
| Publication status | Published - 31 Oct 2023 |
Bibliographical note
Funding Information:The authors would like to thank Joseph Cho for fruitful discussions during an impromptu stay in Kobe that sparked many ideas in this paper. We also express our gratitude to Udo Hertrich-Jeromin for his input which added significant results. Furthermore, we gratefully acknowledge financial support from the FWF research project P28427-N35 “Non-rigidity and symmetry breaking” and the JSPS/FWF Joint Project I3809-N32 “Geometric shape generation”. The first author was also supported by the GNSAGA of INdAM and the MIUR grant “Dipartimenti di Eccellenza” 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino and gratefully acknowledges support from the JSPS Grant-in-Aid for JSPS fellows 19J10679. The third author was partly supported by JSPS KAKENHI Grant Numbers JP18H04489, JP19J02034, JP20K14314, JP20H01801, JP20K03585, JST CREST Grant Number JPMJCR1911, and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). Finally, the authors would like to thank the anonymous referees for their careful reading and insightful suggestions.
Funding
The authors would like to thank Joseph Cho for fruitful discussions during an impromptu stay in Kobe that sparked many ideas in this paper. We also express our gratitude to Udo Hertrich-Jeromin for his input which added significant results. Furthermore, we gratefully acknowledge financial support from the FWF research project P28427-N35 “Non-rigidity and symmetry breaking” and the JSPS/FWF Joint Project I3809-N32 “Geometric shape generation”. The first author was also supported by the GNSAGA of INdAM and the MIUR grant “Dipartimenti di Eccellenza” 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino and gratefully acknowledges support from the JSPS Grant-in-Aid for JSPS fellows 19J10679. The third author was partly supported by JSPS KAKENHI Grant Numbers JP18H04489, JP19J02034, JP20K14314, JP20H01801, JP20K03585, JST CREST Grant Number JPMJCR1911, and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). Finally, the authors would like to thank the anonymous referees for their careful reading and insightful suggestions.
Keywords
- Discrete curvature theory
- Discrete isothermic sphere congruence
- Discrete Omega surface
- Laguerre geometry
- Weierstrass representation
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics