by   Martin Kovár, Alena Chernikava

The properties of quadratic surfaces are important in many different topics of natural sciences, like differential geometry or theoretical physics, but also have many possible applications in engineering. For instance, the algebraic study of a symmetric tensor as a quadratic surface was first introduced already by R. Penrose in 1972. Similar approach was later used also by R. F. Crade, G. S. Hall in their papers on general relativity. Similar mathematical objects are studied from a variety of points of view in many different, recent scientific applications. Among others, we may mention the study of the properties of the causal curves in various models of spacetime which is connected with solving certain systems of differential equations on differentiable manifolds. Therefore, the modeling applications which can display and analyze various types of differentiable surfaces constitute very important tools of study. Our application Quadric Surface Explorer allows to display a three-dimensional live animation of any quadratic surface embedded into the three-dimensional Euclidean space. It calculates its diagonal form using the properties of the eigen-vector spaces, related to the Hermitian matrix associated with the quadric. It also calculates the matrices of the coordinate transformations between the usual cartesian coordinates and the new coordinate system, in which the associated Hermitian matrix get the diagonal form, and displays the principal axes of the given quadratic surface.