# by   Martin Kovár, Alena Chernikava

Abstract and usage:

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.

Input and control:
 Enter the matrix of quadratic coefficients: Hermitian matrix? Enter the vector of linear coefficients: Enter the aditive constant: Equation of the quadric:

Results:

The orthogonal projection of the studied function:

 Matrix of the orthogonal similarity transformation: Diagonal form of the matrix of the quadratic coefficients: Coordinate transformation: Equation of the quadric after the coordinate transformation: Equation of the quadric, final form: The center of the quadric in the new coordinates: The center of the quadric in the original coordinates:

Three-dimensional, live-animated graph of the quadratic surface:

Please, use Shift and Ctrl keys in combination with mouse to modify the view.

The application Quadric Surface Explorer is written in Java powered by Wolfram webMathematica 3.0. The application is hosted at the server of the Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology. For research and scientific activities the software is available free of charge. In all other cases, please contact RNDr. M. Novák, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, phone: +420 5 4114 3135. Acknowledgement: FEKT-S-11-2/921 "Vlastnosti řešení funkcionálních diferenciálních a diferenčních rovnic" 