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MATHEMATICS
Metric invariants of a second-order hypersurface in an $n$-dimensional Euclidean space
D. Yu. Volkova, K. V. Galunovab a St Petersburg State University of Aerospace Instrumentation, 67, ul. Bolshaya Morskaya, St Petersburg, 190000, Russian Federation
b Peter the Great St Petersburg Polytechnic University, 29, ul. Polytechnicheskaya, StPetersburg, 195251, Russian Federation
Abstract:
The article is devoted to the classical problem of analytic geometry in n-dimensional Euclidean space: finding the canonical equation of a quadric. The canonical equation is determined by the invariants of the second-order surface equation. Invariants are quantities that do not change under an affine change of space coordinates. S. L. Pevsner found a convenient system of the following invariants: q is 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 matrix of quadratic terms of the surface equation, i. e. the eigenvalues of this matrix; $K_q$ is the coefficient of the variable $\lambda$ to the power of $n-q$ in a polynomial equal to the determinant of the $n + 1$ order matrix obtained by a certain rule from the original surface equation. All the coefficients of the canonical equations of quadrics are expressed through eigenvalues of the matrix of quadratic terms and the coefficient $K_q$. Pevsner's result is proved in a new way. Elementary properties of determinants are used in the proof. This algorithm for finding the canonical equation of a quadric is a very convenient algorithm for computer graphics.
Keywords:
invariant, second-order hypersurfaces.
Received: 04.10.2021 Revised: 30.11.2021 Accepted: 02.12.2021
Citation:
D. Yu. Volkov, K. V. Galunova, “Metric invariants of a second-order hypersurface in an $n$-dimensional Euclidean space”, Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9:2 (2022), 219–228
Linking options:
https://www.mathnet.ru/eng/vspua4 https://www.mathnet.ru/eng/vspua/v9/i2/p219
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Abstract page: | 88 | Full-text PDF : | 262 | References: | 21 |
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