Metric invariants of a second-order hypersurface in an n-dimensional Euclidean space
DOI:
https://doi.org/10.21638/spbu01.2022.204Abstract
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 λ 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
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References
Литература
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Published
2022-07-06
How to Cite
Volkov, D. Y., & Galunova, K. V. (2022). Metric invariants of a second-order hypersurface in an n-dimensional Euclidean space. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(2), 219–228. https://doi.org/10.21638/spbu01.2022.204
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
