Canonization of the hypersurfaces of the first and the second degree and application for the maximal absolute and relative inaccuracies

In this paper we reduce a linear homogeneous form of n unknowns with an orthogonal transformation of the unknowns in a linear homogeneous form of one unknown with an exactly definite positive coefficient. As an application of this result we find the canonical form of an arbitrary hyperplane in the real n-dimensional affine Euclidean space En. This method for the obtaining of the canonical forms of the hyperplanes is new, since until nowa canonical form of hyperplanes is not defined and is not considered.Besides we find an effective canonical form of the surfaces of the second degree in . Our methodfor the canonization of the surfaces of the second degree in is effective since it gives the exact coefficients of the canonical form of the surface in a dependent ofthe coefficients of the givensurface equation. As an application we give a canonization of the hypersurfaces of the maximal absolute and relative inaccuracies (errors). Besides this method is different from the known approach in the case of the obtaining of a cylinder.