Accurate visualization of graphs of functions of two real variables

The study of a real function of two real variables can be supported by visualization using a Computer Algebra System (CAS). One type of constraints of the system is due to the algorithms implemented, yielding continuous approximations of the given function by interpolation. This often masks discontinuities of the function and can provide strange plots, not compatible with the mathematics. In recent years, point based geometry has gained increasing attention as an alternative surface representation, both for efficient rendering and for flexible geometry processing of complex surfaces. In this paper we present different artifacts created by mesh surfaces near discontinuities and propose a point based method that controls and reduces these artifacts. A least squares penalty method for an automatic generation of the mesh that controls the behavior of the chosen function is presented. The special feature of this method is the ability to improve the accuracy of the surface visualization near a set of interior points where the function may be discontinuous. The present method is formulated as a minimax problem and the non uniform mesh is generated using an iterative algorithm. Results show that for large poorly conditioned matrices, the new algorithm gives more accurate results than the classical preconditioned conjugate algorithm. Keywords—Function singularities, mesh generation, point allocation, visualization, collocation least squares method, Augmented Lagrangian method, Uzawa’s Algorithm, Preconditioned Conjugate Gradient I. GENERAL FRAME OF STUDY With the introduction of the computer into the learning environment the way Mathematics is conveyed has changed. The traditional sequence DefinitionTheorem-Proof has received a complement with examples where visualization plays an important role. This is true in numerous mathematical domains, such as Geometry, using the so-called Dynamical Geometry Packages, and also Analysis and Algebra with Computer Algebra Systems (CAS). Questions about the study of functions and curve discussion have been studied by [20], [11], [12] and others. In particular, various limitations of the usage of the computer have been discussed. Many educators have replaced the traditional sequence mentioned above by another one, beginning with computer assisted experimentation. A well known case of CAS experimentation prior to theoretical study is curve discussion. It happens that a discontinuity of the function or the non-existence of a limit at some point are not shown by the display, despite the fact that a proof is easy to write. Such a cognitive conflict can appear, in a stronger form, * E.S.G. Ecole Supérieure de Gestion 11 Rue Amboise Paré; 75015 Paris; France ** Department of Mathematics -Jerusalem College of Technology Havaad Haleumi Street, 21 -Jerusalem 91160 Israel e-mail: [email protected] [email protected] when studying functions of two real variables. In order for the eye to catch the situation, most Computer Algebra Systems enable a dynamical point of view, using the mouse to turn the surface around. This can help to discover discontinuities or points where partial derivatives cannot exist, but how to be confident of the exactness of the visual impression? As we will see, discontinuities can be hidden. Functions of two real variables are generally introduced in an Advanced Calculus course, where the students discover generalizations of notions learnt in their first Calculus course. The respective roles of the first and second derivatives are extended in the new frame to discover extrema, saddle points and points of inflection. When arriving at the visualization stage, drawings are harder to obtain by hand-work and computerized help is welcome. Plots are obtained via commands similar to the 2Dcommands and the student is confident that what is seen is what has to be seen. We can illustrate the first problem encountered with the example of a paraboloid: when drawn on the black/white board it looks as in Figure 1(a), but generally a CAS plots something like (b) or (c), because of a bounding box. Even when the bounding box is no displayed, it cuts the paraboloid in a visible way.