Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles

Alexander I. Bobenko; Helmut Pottmann; Thilo Rörig

doi:10.1007/s00454-019-00101-1

Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles

Alexander I. Bobenko, Helmut Pottmann, Thilo Rörig

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q∗-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.

Original language	English (US)
Journal	Discrete and Computational Geometry
DOIs	https://doi.org/10.1007/s00454-019-00101-1
State	Published - May 28 2019

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: We would like to thank Wolfgang Schief and Jan Techter for many fruitful discussions. This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics.” – www.discretization.de. Helmut Pottmann was supported through Grant I 2978-N35 of the Austrian Science Fund (FWF).

Access to Document

10.1007/s00454-019-00101-1

https://repository.kaust.edu.sa/bitstream/10754/656384/1/Articlefile1.pdf

Cite this

@article{361a859b3e5e4a32b4199ae78468e50a,

title = "Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles",

abstract = "We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q∗-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.",

author = "Bobenko, {Alexander I.} and Helmut Pottmann and Thilo R{\"o}rig",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: We would like to thank Wolfgang Schief and Jan Techter for many fruitful discussions. This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics.” – www.discretization.de. Helmut Pottmann was supported through Grant I 2978-N35 of the Austrian Science Fund (FWF).",

year = "2019",

month = may,

day = "28",

doi = "10.1007/s00454-019-00101-1",

language = "English (US)",

journal = "Discrete and Computational Geometry",

issn = "0179-5376",

publisher = "Springer Verlag",

}

TY - JOUR

T1 - Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles

AU - Bobenko, Alexander I.

AU - Pottmann, Helmut

AU - Rörig, Thilo

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: We would like to thank Wolfgang Schief and Jan Techter for many fruitful discussions. This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics.” – www.discretization.de. Helmut Pottmann was supported through Grant I 2978-N35 of the Austrian Science Fund (FWF).

PY - 2019/5/28

Y1 - 2019/5/28

N2 - We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q∗-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.

AB - We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q∗-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.

UR - http://hdl.handle.net/10754/656384

UR - http://link.springer.com/10.1007/s00454-019-00101-1

UR - http://www.scopus.com/inward/record.url?scp=85066458569&partnerID=8YFLogxK

U2 - 10.1007/s00454-019-00101-1

DO - 10.1007/s00454-019-00101-1

M3 - Article

SN - 0179-5376

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

ER -

Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this