Classificação de cônicas e quádricas

Abstract

In this paper we perform a study of bilinear forms and quadratic forms in order to classify conic and quadric by their equations, when these curves and surfaces have inclined axes in relation to the cartesian axes. In this cases, when these curves and surfaces are presented in this way their classification does not it's so simple. For this reason, we present the definitions of bilinear forms and quadratic forms, as well as some examples and important results that will help us in the classification process. In this sense, we present the equations of each curve and surface that we are interested in classifying and we presenting applications that can be made with them in our daily life. Then, from the study of quadratic forms, we show that the conic and quadric equations involve a quadratic form, a linear form, and a constant term. In this way, these equations can be placed in matrix form, where one of the matrices involved is the matrix of the quadratic form, which can always be diagonalizable. An orthonormal basis of eigenvectors of this matrix of the quadratic form takes us to the change of coordinates that we wish for the identification of the conic or the quadric in question, since the equation with a new reference makes it easier to classify.