Recognition of computationally constructed loci

Peter Lebmeir; Jürgen Richter-Gebert

doi:10.1007/978-3-540-77356-6_4

Recognition of computationally constructed loci

Peter Lebmeir
, Jürgen Richter-Gebert

Chair of Geometry and Visualization

Technical University of Munich

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

6 Scopus citations

Abstract

We propose an algorithm for automated recognition of computationally constructed curves and discuss several aspects of the recognition problem. Recognizing loci means determining a single implicit polynomial equation and geometric invariants, characterizing an algebraic curve which is given by a discrete set of sample points. Starting with these discrete samples, arising for example from a geometric ruler and compass construction, an eigenvalue analysis of a matrix derived from the data leads to proposed curve parameters. Utilizing the construction itself, with its free and dependent geometric elements, further specifications of the type of constructed curves under genericity assumptions are made. This is done by a second eigenvalue analysis of parameters of several generically generated curves.

Original language	English
Title of host publication	Automated Deduction in Geometry - 6th International Workshop, ADG 2006, Revised Papers
Publisher	Springer Verlag
Pages	52-67
Number of pages	16
ISBN (Print)	354077355X, 9783540773559
DOIs	https://doi.org/10.1007/978-3-540-77356-6_4
State	Published - 2007
Event	6th International Workshop on Automated Deduction in Geometry, ADG 2006 - Pontevedra, Spain Duration: 31 Aug 2006 → 2 Sep 2006

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	4869 LNAI
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	6th International Workshop on Automated Deduction in Geometry, ADG 2006
Country/Territory	Spain
City	Pontevedra
Period	31/08/06 → 2/09/06

Access to Document

10.1007/978-3-540-77356-6_4

Cite this

Lebmeir, P., & Richter-Gebert, J. (2007). Recognition of computationally constructed loci. In Automated Deduction in Geometry - 6th International Workshop, ADG 2006, Revised Papers (pp. 52-67). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4869 LNAI). Springer Verlag. https://doi.org/10.1007/978-3-540-77356-6_4

@inproceedings{11b91dd2ff39442ea09a6d61dc4f470c,

title = "Recognition of computationally constructed loci",

abstract = "We propose an algorithm for automated recognition of computationally constructed curves and discuss several aspects of the recognition problem. Recognizing loci means determining a single implicit polynomial equation and geometric invariants, characterizing an algebraic curve which is given by a discrete set of sample points. Starting with these discrete samples, arising for example from a geometric ruler and compass construction, an eigenvalue analysis of a matrix derived from the data leads to proposed curve parameters. Utilizing the construction itself, with its free and dependent geometric elements, further specifications of the type of constructed curves under genericity assumptions are made. This is done by a second eigenvalue analysis of parameters of several generically generated curves.",

author = "Peter Lebmeir and J{\"u}rgen Richter-Gebert",

year = "2007",

doi = "10.1007/978-3-540-77356-6\_4",

language = "English",

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series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

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booktitle = "Automated Deduction in Geometry - 6th International Workshop, ADG 2006, Revised Papers",

note = "6th International Workshop on Automated Deduction in Geometry, ADG 2006 ; Conference date: 31-08-2006 Through 02-09-2006",

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Lebmeir, P & Richter-Gebert, J 2007, Recognition of computationally constructed loci. in Automated Deduction in Geometry - 6th International Workshop, ADG 2006, Revised Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4869 LNAI, Springer Verlag, pp. 52-67, 6th International Workshop on Automated Deduction in Geometry, ADG 2006, Pontevedra, Spain, 31/08/06. https://doi.org/10.1007/978-3-540-77356-6_4

Recognition of computationally constructed loci. / Lebmeir, Peter; Richter-Gebert, Jürgen.
Automated Deduction in Geometry - 6th International Workshop, ADG 2006, Revised Papers. Springer Verlag, 2007. p. 52-67 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4869 LNAI).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - We propose an algorithm for automated recognition of computationally constructed curves and discuss several aspects of the recognition problem. Recognizing loci means determining a single implicit polynomial equation and geometric invariants, characterizing an algebraic curve which is given by a discrete set of sample points. Starting with these discrete samples, arising for example from a geometric ruler and compass construction, an eigenvalue analysis of a matrix derived from the data leads to proposed curve parameters. Utilizing the construction itself, with its free and dependent geometric elements, further specifications of the type of constructed curves under genericity assumptions are made. This is done by a second eigenvalue analysis of parameters of several generically generated curves.

AB - We propose an algorithm for automated recognition of computationally constructed curves and discuss several aspects of the recognition problem. Recognizing loci means determining a single implicit polynomial equation and geometric invariants, characterizing an algebraic curve which is given by a discrete set of sample points. Starting with these discrete samples, arising for example from a geometric ruler and compass construction, an eigenvalue analysis of a matrix derived from the data leads to proposed curve parameters. Utilizing the construction itself, with its free and dependent geometric elements, further specifications of the type of constructed curves under genericity assumptions are made. This is done by a second eigenvalue analysis of parameters of several generically generated curves.

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DO - 10.1007/978-3-540-77356-6_4

M3 - Conference contribution

AN - SCOPUS:38549168899

SN - 354077355X

SN - 9783540773559

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

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BT - Automated Deduction in Geometry - 6th International Workshop, ADG 2006, Revised Papers

PB - Springer Verlag

T2 - 6th International Workshop on Automated Deduction in Geometry, ADG 2006

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ER -

Lebmeir P, Richter-Gebert J. Recognition of computationally constructed loci. In Automated Deduction in Geometry - 6th International Workshop, ADG 2006, Revised Papers. Springer Verlag. 2007. p. 52-67. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-540-77356-6_4

Recognition of computationally constructed loci

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