COGNITIVE COMPUTER GRAPHICS AS A MEANS OF "SOFT" MODELING IN PROBLEMS OF RESTORATION OF FUNCTIONS OF TWO VARIABLES - Khomchenko, А. Н.; Koval, Н. В.; Osipova, Н.В. (Journal of Information Technologies in Education (ITE), 2016)
The paper considers the problem of bi-cubic interpolation on the final element of serendipity family. With cognitive-graphical analysis the rigid model of Ergatoudis, Irons and Zenkevich (1968) compared with alternative models, obtained by the methods: direct geometric design, a weighted averaging of the basis polynomials, systematic generation of bases (advanced Taylor procedure). The emphasis is placed on the phenomenon of "gravitational repulsion" (Zenkevich paradox). The causes of rising of inadequate physical spectra nodal loads on serendipity elements of higher orders are investigated. Soft modeling allows us to build a lot of serendipity elements of bicubic interpolation, and you do not even need to know the exact form of the rigid model. The different interpretations of integral characteristics of the basis polynomials: geometrical, physical,probability are offered.Under the soft model in the theory of interpolation of function of two variables implies the model amenable to change through the choice of basis. Such changes in the family of Lagrangian finite elements of higher orders are excluded (hard simulation). Standard models of serendipity family (Zenkevich) were also tough. It was found that the "responsibility" for the rigidity of serendipity model rests on ruled surfaces (zero Gaussian curvature) - conoids that predominate in the base set. Cognitive portraits zero lines of standard serendipity surfaces suggested that in order to "mitigate" of serendipity pattern conoid should better be replaced by surfaces of alternating Gaussian curvature. The article shows the alternative (soft) bases of serendipity models. The work is devoted to solving scientific and technological problems aimed at the creation, dissemination and use of cognitive computer graphics in teaching and learning. The results are of interest to students of specialties: "Computer Science and Information Technologies", "System Analysis", "Software Engineering", as well as post-graduate specialty "Information Technologies".