Metric invariants of a second-order hypersurface in an n-dimensional Euclidean space

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 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.