{"title":"Metric Invariants of Second-Order Surfaces","authors":"D. Yu. Volkov, K. V. Galunova","doi":"10.1134/s1063454123040210","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper is devoted to the classical problem of analytical geometry in <i>n</i>-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: <i>q</i>, 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 <i>K</i><sub><i>q</i></sub>, the coefficient of the variable λ to the power <i>n</i> – <i>q</i> in the polynomial that is equal to the determinant of the matrix of order <i>n</i> + 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 <i>K</i><sub><i>q</i></sub> 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.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"121 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik St Petersburg University-Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1063454123040210","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍：
Vestnik St. Petersburg University, Mathematics is a journal that publishes original contributions in all areas of fundamental and applied mathematics. It is the prime outlet for the findings of scientists from the Faculty of Mathematics and Mechanics of St. Petersburg State University. Articles of the journal cover the major areas of fundamental and applied mathematics. The following are the main subject headings: Mathematical Analysis; Higher Algebra and Numbers Theory; Higher Geometry; Differential Equations; Mathematical Physics; Computational Mathematics and Numerical Analysis; Statistical Simulation; Theoretical Cybernetics; Game Theory; Operations Research; Theory of Probability and Mathematical Statistics, and Mathematical Problems of Mechanics and Astronomy.
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