Metric Invariants of Second-Order Surfaces

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 K_q, 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_q 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.