Metric Invariants of Second-Order Surfaces.

Λεπτομέρειες βιβλιογραφικής εγγραφής
Τίτλος:	Metric Invariants of Second-Order Surfaces.
Συγγραφείς:	Volkov, D. Yu., Galunova, K. V.
Πηγή:	Vestnik St. Petersburg University: Mathematics; Dec2023, Vol. 56 Issue 4, p569-576, 8p
Περίληψη:	The paper is devoted to the classical problem of analytical geometry in n-dimensional Euclidean space, namely, finding the canonical equation of a quadric from an initial equation. The canonical equation is determined by the invariants of the second-order surface equation, i.e., by quantities that do not change when the space coordinates are changed affinely. S.L. Pevzner found a convenient system containing the following invariants: q, the rank of the extended matrix of the system for determining the center of symmetry of the surface; the roots of the characteristic polynomial of the quadratic term matrix of the surface equation, i.e., the eigenvalues of this matrix; and Kq, the coefficient of the variable λ to the power n – q in the polynomial that is equal to the determinant of the matrix of order n + 1 that is obtained according to a certain rule from the initial surface equation. The eigenvalues of the matrix of the quadratic terms and coefficient Kq make it possible to write the canonical equation of the surface. In the paper, we propose a new simple proof of Pevzner's result. In the proof, only elementary properties of the determinants are used. This algorithm for finding the canonical surface equation can find application in computer graphics. [ABSTRACT FROM AUTHOR]
	Copyright of Vestnik St. Petersburg University: Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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Header	DbId: edb DbLabel: Complementary Index An: 175005063 RelevancyScore: 944 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 943.786315917969
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Items	– Name: Title Label: Title Group: Ti Data: Metric Invariants of Second-Order Surfaces. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Volkov%2C+D%2E+Yu%2E%22">Volkov, D. Yu.</searchLink><br /><searchLink fieldCode="AR" term="%22Galunova%2C+K%2E+V%2E%22">Galunova, K. V.</searchLink> – Name: TitleSource Label: Source Group: Src Data: Vestnik St. Petersburg University: Mathematics; Dec2023, Vol. 56 Issue 4, p569-576, 8p – Name: Abstract Label: Abstract Group: Ab Data: The paper is devoted to the classical problem of analytical geometry in n-dimensional Euclidean space, namely, finding the canonical equation of a quadric from an initial equation. The canonical equation is determined by the invariants of the second-order surface equation, i.e., by quantities that do not change when the space coordinates are changed affinely. S.L. Pevzner found a convenient system containing the following invariants: q, the rank of the extended matrix of the system for determining the center of symmetry of the surface; the roots of the characteristic polynomial of the quadratic term matrix of the surface equation, i.e., the eigenvalues of this matrix; and K<subscript>q</subscript>, the coefficient of the variable λ to the power n – q in the polynomial that is equal to the determinant of the matrix of order n + 1 that is obtained according to a certain rule from the initial surface equation. The eigenvalues of the matrix of the quadratic terms and coefficient K<subscript>q</subscript> make it possible to write the canonical equation of the surface. In the paper, we propose a new simple proof of Pevzner's result. In the proof, only elementary properties of the determinants are used. This algorithm for finding the canonical surface equation can find application in computer graphics. [ABSTRACT FROM AUTHOR] – Name: Abstract Label: Group: Ab Data: <i>Copyright of Vestnik St. Petersburg University: Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo	BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1134/S1063454123040210 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 8 StartPage: 569 Titles: – TitleFull: Metric Invariants of Second-Order Surfaces. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Volkov, D. Yu. – PersonEntity: Name: NameFull: Galunova, K. V. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 12 Text: Dec2023 Type: published Y: 2023 Identifiers: – Type: issn-print Value: 10634541 Numbering: – Type: volume Value: 56 – Type: issue Value: 4 Titles: – TitleFull: Vestnik St. Petersburg University: Mathematics Type: main
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