The need to improve the level of mathematical in particular geometric training of students of technical universities is due to modern technologies of computer-aided design. They are based on mathematical models of designed products, technological processes, etc., taking into account a large variety of source data. Therefore, from the first years of technical universities, when studying the cycle of mathematical disciplines, it is advisable to interpret a number of issues in terms and concepts of multidimensional geometry. At the same time, the combination of constructive (graphical) algorithms for solving problems in descriptive geometry with analytical algorithms in linear algebra and matanalysis allows us to summarize their advantages: the constructive approach provides the imagery inherent in engineering thinking, and the analytical approach provides the final result. The article shows the effectiveness of combining constructive and analytical algorithms for solving problems involving linear and nonlinear forms of many variables using specific examples.
In the first years study of technical universities two approaches are considered when studying mathematical cycle disciplines: − constructive (graphic) in the teaching of descriptive geometry; − analytical with emphasis on the study of numerical algorithms (linear algebra, calculus).
Only in the course of analytical geometry are algebra and geometry considered together. At the same time, none of these courses even talk about multidimensional spaces, although they consider systems of linear equations from several unknowns, study methods for differentiating and integrating functions of many variables ets. Each of these approaches has its own advantages. If the constructive approach provides the imagery inherent in engineering thinking, then the analytical approach provides the final result. Therefore, their rational combination should contribute to the successful development of the course being studied. In this regard, this publication is devoted to the justification of making some additions to the course of descriptive geometry, which, in our opinion, will contribute to the geometric support of algorithms for solving a number of problems of higher mathematics.
Let's start with linear algebra. In high school, students are taught to solve systems of two linear equations with two unknowns and three linear equations with three unknowns. Students understand the geometric meaning of the systems being solved. In the first case, they calculate the coordinates of the intersection point of two straight lines, and in the second – the coordinates of the common point of three planes. At the University, they study the solution of systems of four or more linear equations with the corresponding number of unknowns, using the Gauss algorithm. Unfortunately, now them do not explain the geometric meaning of a linear equation from many unknown ones, because the programs of existing courses in descriptive and analytical geometry are focused only on the study of linear and nonlinear forms of threedimensional space.
This gap can be most easily and clearly eliminated by expanding the subject of descriptive geometry with the forms of four-dimensional space and generalizing the twocard drawing of Monge with the drawing of Radishchev (Fig. 1).
At the same time, it is logical and simple enough to
graphical definition lines and planes of three-dimensional
space ([
all the hyperplanes of this n dimensional space pass.
Thus, the extension of the subject of descriptive geometry by multidimensional (at the first stage − fourdimensional ) linear forms and their analytical assignment in the form of linear equations or systems of equations allows us to visually (geometrically figuratively) represent them
as p - planes, their intersections and unions (enclosing spaces). As a result, the existence of a relationship between their methods and the rationalization of algorithms for solving certain problems is revealed.
To confirm this thesis, section 6.2.1 [
In the language of linear algebra this problem is reduced to solving a system of three linear equations with three unknowns: by elementary transformations, the square matrix of coefficients is reduced to a trapezoidal one , which in descriptive geometry corresponds to the transformation of one plane into a projecting one.
Therefore, a simpler calculation of the determinant of the transformed (trapezoidal) matrix corresponds to a simpler graphical way of constructing a common point of three planes, one of which is the projecting one.
The
second
example
shows
a
graphical
implementation of the Gauss method for sequentially
reducing the dimension of the problem to be solved ([
3 (ABSD), the drawing clearly shows a sequential decrease in the dimension 4→3→2→1.
Thus, the questions discussed above convincingly show the unity of the subject of linear algebra and multidimensional descriptive geometry, the usefulness of parallel solutions of geometric problems using graphical and analytical methods.
Let's consider examples of geometric support for solving problems involving nonlinear forms. Therefore, we will first show the kinematic method of their formation, which is characterized by clarity and implements the principle of separation, which is widely used in computational mathematics.
In the Oxy coordinate plane, point A, moving according
to some law, forms a curve 1(y = f(x)) (fig. 2). In turn, the
curve 1, moving in the space Oxyz by its law, forms
("sweeps") a two- dimensional surface 2(z = γ (x, y)). The
surface 2, moving in the four- dimensional space Oxyzt
in the direction of the axis Ot, forms a 3-surface 3 (t =
φ(x, y, z)), which, in turn, "sweeps" the 4-surface 4 (in
fig.2 not shown). This process continues until the (n – 2)
surface −2 "sweeps" the hypersurface −1.
Thus, the hypersurface −1 is a one - parameter ∞
- dimensional nonlinear flag (see [
The question arises, how to construct a tangent plane to the hypersurface being constructed?
In the course of mathematical analysis, the tangent t to the curve m at its point Μ is called the limit position of the secant ΜΝ, which it occupies when the point N along the curve t tends to the point Μ. In other words, a tangent t is such a secant (chord) that intersects the curve m at two coinciding points = . This definition also applies to the touch of curved lines, flat and spatial. Since two curves can intersect at several points, two, three, or more points can coincide in the limit. Therefore, they talk about two-point, three-point, etc. touches. For example, two second-order curves can have two-point, three-point, and four-point touches. In the language of mathematical analysis, this means that in the case of a two-point touch, the coordinates of the coinciding points satisfy the equations of both curves, and the first derivatives taken from the equations of these curves are equal at this point. In the case of a three-point touch, the second derivatives are additionally equal, and in the case of a four-point touch, the third derivatives are also equal. In engineering practice, it is customary to call two-point, three-point, etc. touches, respectively, touches of the first, second, and n-th order of smoothness. Curves made up of arcs that touch curves are called outlines.
These concepts are applied in the appropriate interpretation to the touch of surfaces. As noted above, the construction of curves and surfaces is of great practical importance. In theory, they are generalized to touch in multidimensional spaces. In computational terms, this is reduced to operating with partial derivatives of functions of many variables.
Let's start by considering the theoretical provisions for constructing a tangent plane to a surface in threedimensional space.
In differential geometry, it is shown that the set of
tangents drawn to a surface Φ at some point A belongs
to the plane τ, if the point A is its regular (ordinary) point.
If the point A is a special point of the surface Φ, then the
set of tangents forms a conic surface τ with a vertex at
this point [
Since the tangent plane τ is uniquely defined by two straight lines, the algorithm for constructing it consists of the following steps:
- through this point A of the surface Φ, any two of its lines a, b are drawn;
- at point A, tangents 1, 2 are constructed to the selected lines a, b; intersecting lines 1, 2 define the plane τ, touching the surface Φ at point A.
This algorithm is the basis of an analytical method for constructing a tangent plane τ of a surface Φ at its point A. If the equation Φ (x, y, z) = 0 of the surface is substituted with the values = , = , = , then we get the equations of the sections a, b, with the surface Φ by planes passing through the point A and parallel to the coordinate planes Oyz, Oxz, Oxy, respectively. Partial derivatives Φ( , , ), Φ( , , ) , Φ( , , )
at point A( , , ) are the angular coefficients of the tangents 1, 2, 3, held at point A for curves a, b, c.
The equation of the tangent plane τ has the form: Φ Φ Φ
Thus, analytically, the construction of a tangent plane in three-dimensional space is reduced to the calculation of partial derivatives of functions Φ (x, y, z) = 0 from three variables.
( − ) + ( − ) + ( − ) = 0
Let's consider an example of constructing a tangent plane 2 to a surface Φ2 in three-dimensional space. In textbooks on descriptive geometry, the construction of tangent planes to the simplest surfaces (sphere, cone, etc.) is given. On the surface at this point A we draw two graphically simple lines a and b.
The tangents and define the desired tangent plane τ∋ A. Since engineering surfaces are complex, the curves a and b take the surface sections as planes parallel to the coordinate planes of the projections. In our example, the tangent plane 2 is structurally defined by two tangents 1, 1, drawn to the sections 1, 1 of the given surface Φ2 (Fig. 3).
Structurally, the method of stratification is used to
solve such problems. The application of the bundle idea in
solving problems involving nonlinear forms is shown in
[
We generalize the solution of this problem to the construction of a tangent of the 3 - plane 3 to the 3-surface Φ3 in four-dimensional space.
In the fig. 4 shows an example of generalization of the considered algorithm to the construction of a tangent of the 3 - plane 3 to the 3-surface Φ3 in four-dimensional space.
u = f (x, y, z).
Using the three known coordinates , , of a certain point A, we calculate its fourth coordinate from this equation. Structurally, this is done by drawing three projecting 3-planes Г3 (x = ), Г3 (y = ), Г3 (z = ). Each of them intersects this 3-surface Φ3, respectively, on 2-surfaces (3 + 3 – 4 = 2): 2 = Γ 3 ∩ Φ3, 2 = Γ 3 ∩ Φ3 2 = Γ 3 ∩ Φ3 (they are not shown in fig. 4).
These three 2-surfaces belonging to the 3-surface of Φ3 intersect in pairs along three flat curves (2 + 2 – 3 = 1), belonging to 2-planes parallel to the corresponding coordinate planes: 1 = 2 ∩ 2 = 1( ) , 1 = 2 ∩ 2 = 2( ) , 1 = 2 ∩ 2 ( = 3( )).
It follows that 1 ∥ , 1 || , 1 || . These three curves intersect at A ∈ Ф3. The angular coefficients of the tangent 1, 1, 1, carried out at the point to these curves, the essence of partial derivatives , , . Three tangents 1, 1, 1 passing through point A and located in perpendicular planes define the desired 3-plane 3, tangent to this 3-surface Φ3 at its point A. Subsequent generalizations of the algorithm are fairly obvious. Such generalizations of the above to higher-dimensional spaces explain the geometric meaning of partial derivatives of functions of n-variables.
The application of the bundle idea in solving problems
involving nonlinear forms is shown in [
Knowledge of the algorithm for solving the problem would allow the student to establish a close connection with other special disciplines at the stage of design, calculations and visualization of data in CAD systems.
The real implementation of departments engineering graphics of the concept of geometric support of algorithms for solving problems of higher mathematics is possible in the educational process only in the presence of highly qualified teachers who possess constructive (graphical) and analytical methods for solving geometric problems. Unfortunately, several generations of teachers of the Department of engineering graphics have considered and continue to consider descriptive geometry as a purely graphical discipline and analytical methods for solving problems do not apply. Therefore, the actual task of the scientific and methodological Council is to organize an effective system of professional development of teachers.