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.

中文翻译：

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二阶曲面的度量不变量

摘要

本文致力于解决n维欧几里得空间中解析几何的经典问题，即从初始方程求二次方程的正则方程。正则方程由二阶表面方程的不变量确定，即由空间坐标仿射改变时不改变的量确定。SL Pevzner 发现了一个方便的系统，包含以下不变量：q，系统扩展矩阵的秩，用于确定表面的对称中心；表面方程的二次项矩阵的特征多项式的根，即该矩阵的特征值；K _q是多项式中变量 λ 的n – q次方的系数，该多项式等于从初始曲面方程按一定规则获得的n + 1 阶矩阵的行列式。二次项矩阵的特征值和系数K _q使得可以写出曲面的规范方程。在本文中，我们提出了 Pevzner 结果的新的简单证明。在证明中，仅使用行列式的基本性质。这种求正则曲面方程的算法可以在计算机图形学中得到应用。