Gaspard Monge wrote: "The charm that accompanies science can overcome man's natural aversion to the mind intenseness and make them find pleasure in their mind’s exercise that for most of people seems as tiresome and boring occupation". He had written it including descriptive geometry. To exercise one’s mind — what is this but the brain building, and science is accompanied just by heuristic thinking, so that brings new discoveries for an intellectual. The most difficult in descriptive geometry is the ability to represent a spatial geometric figure or such figures’ combination on two images. It is clear that the usual problems of a course are resolved within the academic discipline, and are typical ones, readily understandable for any student of a technical high educational institution, while the tasks at Academic Olympics, even if these tasks are destined for use inside a high educational institution, are more difficult. If for a solving of problems from an ordinary problem book on descriptive geometry’s course it is enough to know literally a few algorithms, for tasks of increased difficulty that is not enough. The Academic Olympics’ functions reveal such a feature of those on descriptive geometry as their inseparable property to be a catalyst for development of heuristic thinking. Here there is not only the disclosure of students’ abilities to solve ordinary geometric problems, but the ability to solve problems of heuristic direction in general. It is obvious that knowledge of typical problems on the course of descriptive geometry is absolutely insufficiently, as well as it is insufficiently to know school geometry, that currently almost is not teaching in schools — now it is necessary to have not only the spatial perception, but at least the beginnings of heuristic thinking. This, plus the mobilization of all mental resources, contributes both to the solution of given geometric problems, and further solving other problems in the related areas of science and technology.
Geometric simulation is creation of a geometric model, whose properties and characteristics in a varying degree determine the subject of investigation’s properties and characteristics. The geometric model is a special case of the mathematical model. The feature of the geometric model is that it will always be a geometric figure, and therefore, by its very nature, is visual. If the mathematical model is a set of equations, which says little to an ordinary engineer, the geometric model as representation of the mathematical model and as the geometric figure itself, enables to "see" this set. Any geometric model can be represented graphically. Graphical model of an object is a mapping of its geometric model onto a plane (or other surface). Therefore, the graphical model can be considered as a special case of the geometric model. Graphical models are very various – these are graphics, and graphical structures of immense complexity, reflecting spatial geometric figures. These are drawings of geometric figures, simulating processes of all kinds. The simulation goes on as follows. According to known geometric and differential criteria the geometric model is executed. Then a mathematical model is composed based on the geometric model, finally a computer program is compiled on the mathematical model. As a result of consideration in this paper the process of obtaining the geometric models of surface and linear forms for auto-roads it is possible to make a following conclusion. For geometric simulation and the consequent mathematical one the descriptive geometry involvement is vital. Just the descriptive geometry is used both on the initial and final stages of design.
In the first part of this paper has discussed about the basic properties of cyclide Dupin, and has gave some examples of their applications: three ways of solving the problem of Apollonius exclusively by means of compass and ruler, using identified properties cyclide Dupin, that is, given a classical solution of the problem. In the second of part of the work continued consideration of the properties of cyclide Dupin. Proposed and proved the possibility ask cyclide Dupin arbitrary ellipse as the center line of the forming a plurality of spheres and a sphere with the center belong - ing to the ellipse. Proved the adequacy of this information is used to build the cyclide Dupin. Geometrically proved that the focal line of cychlid are not that other, as curves of the second order. Given the graphical representation of the focal lines of cychlid. Shown polyconic compliance focal lines of cichlid of Dupin, which is considered in all four cases. The proposed formation of the hyperbolic surfaces of the fourth order with one or two primary curves of the second order, in this case ellipses. Apply sofocus this ellipse the hyperbola. Although the primary focus of the ellipse lying in the plane of the hoe, with the center coinciding with the origin of coordinates, is stationary, and the coordinate system rotates around the z axis. Then the points of intersection of the rotating coordinates x and y with a fixed ellipse specify new values for the major and minor axis of the ellipse with resultant changes in the form defocuses of the hyperbola. Although this modeling is not directly connected with Cychlidae Dupin, but clearly follows from the properties of its focal curves – curves of the second order. Withdrawn Equations of the surface and its throat.
Quadratic programming problems are one of special cases of mathematical programming problems. Mathematical programming problems solution is of great importance, because these problems are those of optimizing of solution related to presented issues from multitude of possible ones. The mathematical programming problems are linear, nonlinear, dynamic and others. It is suggested to consider a graph-analytic solution of quadratic programming’s special problems, which, taken together, constitute the quadratic programming problems for two and three variables. A total of eight special problems have been considered.
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