Prof., Doctor of sciences Volkov V.Ya.,

Ph.D. Ilyasova O.B.,

Ph.D. Kaygorodtseva N.V.

Siberian State Automobile and Road Academy, Russian Federation

Improvement of the geometrical graphic preparation of graduates of engineering specialties of technical universities

Now in a number of leading technical universities in Europe and America, as an alternative to the traditional courses of descriptive geometry and engineering graphics, a course of engineering graphics and computer modeling, with some elements of descriptive geometry is proposed; it greatly simplifies the content of these courses, reducing them to elementary geometrical graphic information. We believe it will essentially lower level of geometric, design and creative training of future engineers and further it will affect the level of development and geometric CAD system device of graphic editors.

We think that as a new paradigm of the traditional course of descriptive geometry, it is necessary to do this course more mathematical, introducing the theory of geometric parameterization not only objects but also the geometrical conditions, to formalize the solutions of geometric problems, using an analysis of their conditions and to do the processes of designing of surfaces and different manifolds in a form of the algorithm. At the same time are of interest the generalizations of geometric apparatus on space of different dimensions and structural characteristics. Such an approach, in our opinion, will significantly improve the training of engineering specialists, by developing not only spatial, but also logical thinking; it will also create a geometric apparatus which can be used as a basis for CAD system and engineering software products.

To determine the dimension of the space of images and preimages formulas of counting the dimension of the linear and curvilinear objects from the theory of parametrisation are applied.

As a basic calculation equation for determining the dimension (parametric number) linear object the formula of Hermann Grassmann [7], which is contained in a number of references [5,6] is used:

, (1)

where n - dimension of the space, in which the Grassmann manifold is considered, m - dimension of the plane (element), which forms the Grassmann manifold.

Besides linear elements in the space E3, there are nonlinear, that is, curvilinear elements: algebraic curves and surfaces. Calculation of dimensions of these elements in a space can be done according to the formula of parametric numbers [2]:

, (2)

where m - the order of an algebraic curve, n - the dimension of space.

To determine the dimension of linear combinations of sets of objects the formula is used[2]:

(3)

where ri - the number of basic objects - the dimension of the Grassmann manifold mi - the main objects belonging to the planes (spaces), p - the number of different basic objects.

The dimension of Shuberts’ manifolds of m-planes by the formula [2] is calculated:

(4)

To define the dimension of combinations of Shuberts’ manifolds the formula [2] is used:

(5)

To determine the dimension of sets of linear combinations and curved objects the formula [1] is used:

, (6)

where n - dimension of the ambient space, m - the dimension of the required main object , p - the order of curves and surfaces.

Now, at the time of the digital, information and software technology, when all the information (images, sound, etc.) are encoded in the symbols and signs, particularly relevant method proposed by Schubert [8] which is currently being developed by Professor V. Ya. Volkov [3] .

The method consists in the formal representation of the generalized condition of incidence of special symbols, and the possibility of using special algebraic operations to calculate its dimension.

It is proposed by scientists’ geometry to use the letter e (pronounced "eshka") in order to describe the conditions of incidence and in order to identify appropriate sets, then the generalized condition of incidence will be represented in symbolic form [3]:

(7)

In the following notation the number of upper and lower indices are equal, and the quantities m, m-1, ..., 1, 0; am, am-1, ..., a1, a0 - take the values of natural numbers, including "zero" (0, 1 , 2, 3, 4, ...), ie values of the numbers which are used to denote the number of objects. The values of the upper indices m, m-1, ..., 1, 0, define the dimension of the required linear manifold (the very meaning of m) and the dimensions of all its linear subsets up to the point. The lower indices define the dimension of space or subspace, to which belongs the required element.

A generalized condition of incidence, as well as geometric objects of the space, which are characterized by the dimension, which is determined by the formula [2]:

, (8)

where n - dimension of the space in which the incidence is considered, m - dimension of the space (element), that satisfies the generalized condition of incidence, ai – the lower indices in symbolic interpretations of condition (2).

After calculating the degree of parallelism the dimension of the conditions of parallelism can be determined according to the formula [3]:

, (9)

where p / / - value of the degree of parallelism, n - dimension of the space, in which the condition of parallelism is regarded, m and q - the dimension of parallel elements, and besides .

Before the direct process of graphical modeling the properties of the simulated set can be determined by the properties of prototypes - the model and the specified display, that is, to do an analysis of baseline data and with some calculations, to predict the number of responses and their dimension. But first of all, it is necessary to make sure in the adequacy of the modeled space and in particular in the adequacy of the constructed model.

Here the notion of "adequacy model" is taken as the conventional concept of fit of the model which is modeled as an object or process. In this case we mean adequacy not generally, but according to the properties of models that are considered essential for the study [4].

Three axioms [2] are accepted for an adequate required model:

Axiom 1. The dimension of the space of the images and preimages are identical.

Axiom 2. Structural characteristics of the space of images and preimages are equivalent.

Axiom 3. Space of images and preimages are defined relative to the same class of geometric problems.

On the basis of three axioms, simple graphical models, which can solve the problem of projective, affine and metric spaces, are presented.

The model of the 3-dimensional space:

The projective space

Fig 1. The model of intersecting lines Fig.2 The intersection of the line and

The plane

The affine space: The metric space:

Fig.3 Positional and affine problems Flg.4 Positional, affine and metric

problems (Mongers’ draft)

Innovation transformed descriptive geometry mustn’t be the discipline which serves the engineering graphics and plays the role of the theoretical bases for the construction drawings. It can and should become a mathematical discipline with the evidence base and get right for further development because a solving of the problems of a constructive, analytical and axiomatic modeling in constructing the displays are still actual today and will remain so tomorrow.

References:

1. A. Volkov V.Ya. Graphics optimization models of multivariate processes: a monograph / V.Ya. Volkov, M.A. Chizhik – Omsk, Omsk State Service University, 2009. - 101 p.

2. Volkov, V. Ya. Multivariate enumerative geometry: a monograph / V.Ya. Volkov, V. Yu Yurkov. – Omsk, Omsk State Pedagogical University, 2008. - 244 p.

3. Volkov V.Ya. The theory of parameterization and modeling of geometric objects of multidimensional spaces and its applications: Abstract. Thesis of Doctor of engineering science / VY Volkov. - Moscow Aviation Institute, 1983. - 27 p.

4. Lopatnikov L.I. Economics and Mathematics Dictionary: Dictionary of modern economics / L.I. Lopatnikov. - 5th ed., Revised. and add. - Moscow: Delo, 2003. - 520.

5. Rosenfeld B.A. Multidimensional Space / B.A. Rosenfeld. - Moscow: Nauka, 1966. - 647 p.

6. Chetverukhin N.F. Parameterization and its applications in geometry / N.F. Chetverukhin, L. Jackiewicz / / Mathematics in School, 1963. - № 5. - S. 15-23.

7.Grassmann H. Die lineare Ausdehnungslehre ein neuer Zaweig der Mathematik / H. Grassmann. – Leipzig, 1844. – 279 s.

8.Schubert H. Kalkul der abzahlenden Geometrie / H. Schubert. – Berlin, heidelberg, New-York: Springer Verlag, 1979. – 349 s.