=Paper=
{{Paper
|id=Vol-2763/CPT2020_paper_s7-3
|storemode=property
|title=Geometric support of algorithms for solving Problems of higher mathematics
|pdfUrl=https://ceur-ws.org/Vol-2763/CPT2020_paper_s7-3.pdf
|volume=Vol-2763
|authors=Ilzina Dmitrieva,Gennady Ivanov,Alexey Mineev
}}
==Geometric support of algorithms for solving Problems of higher mathematics==
https://ceur-ws.org/Vol-2763/CPT2020_paper_s7-3.pdf
Geometric support of algorithms for solving Problems of higher
mathematics
I.M. Dmitrieva1, G.S. Ivanov2, A.B. Mineev2
ilzina@yandex.ru | ivanov_gs@rambler.ru | mineev30@yandex.ru
1
Mytischi Branch of Bauman Moscow State Technical University, Mytischi-5, Moscow Region, Russia
2
Bauman Moscow State Technical University, Moscow, Russia
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.
Keywords: descriptive geometry, linear algebra, multidimensional forms - linear and nonlinear, constructive and analytical
solutions, geometric model
the study of linear and nonlinear forms of three-
1. Introduction dimensional space.
In the first years study of technical universities two This gap can be most easily and clearly eliminated by
approaches are considered when studying mathematical expanding the subject of descriptive geometry with the
cycle disciplines: forms of four-dimensional space and generalizing the
− constructive (graphic) in the teaching of descriptive twocard drawing of Monge with the drawing of
geometry; Radishchev (Fig. 1).
− 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.
2. Geometric representation of the solution of
systems of linear equations
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 Fig. 1. Setting point A of a four-dimensional space in the
unknowns. Students understand the geometric meaning of Radishchev drawing
the systems being solved. In the first case, they calculate
the coordinates of the intersection point of two straight At the same time, it is logical and simple enough to
lines, and in the second – the coordinates of the common graphical definition lines and planes of three-dimensional
point of three planes. At the University, they study the space ([1]: 2.1, 2.2) and generalize it to define linear forms
solution of systems of four or more linear equations with of multidimensional space ([1]: 2.2.3). This fully applies
the corresponding number of unknowns, using the Gauss to their analytical task ([1]: 2.3). If linear algebra courses
algorithm. Unfortunately, now them do not explain the do not provide a geometric interpretation of a rectangular
geometric meaning of a linear equation from many matrix and its rank , this is now available ([1]: 2.3.2):
unknown ones, because the programs of existing courses
in descriptive and analytical geometry are focused only on
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY
4.0)
− a rectangular matrix consisting of n-p rows and n + 1 graphical way of constructing a common point of three
columns defines a p-plane defined by a system of n-p planes, one of which is the projecting one.
linear equations from n unknowns; The second example shows a graphical
− rank R = n-p, where p is the dimension of the p-plane implementation of the Gauss method for sequentially
through which all the hyperplanes of this n - reducing the dimension of the problem to be solved ([1],
dimensional space pass. p. 6.2.1). On the example of a graphical solution to the
Thus, the extension of the subject of descriptive problem of constructing a point K of the intersection of a
geometry by multidimensional (at the first stage − four- line l with a hyperplane Σ 3 (ABSD), the drawing clearly
dimensional ) linear forms and their analytical assignment shows a sequential decrease in the dimension 4→3→2→1.
in the form of linear equations or systems of equations Thus, the questions discussed above convincingly
allows us to visually (geometrically figuratively) represent show the unity of the subject of linear algebra and
them as p - planes, their intersections and unions multidimensional descriptive geometry, the usefulness of
(enclosing spaces). As a result, the existence of a parallel solutions of geometric problems using graphical
relationship between their methods and the rationalization and analytical methods.
of algorithms for solving certain problems is revealed.
To confirm this thesis, section 6.2.1 [1] provides two 3. Kinematic method for forming
examples. In the first example, we discuss an algorithm for multidimensional surfaces
constructing the intersection point of three planes α, β, and Let's consider examples of geometric support for
γ. As a rule, in descriptive geometry, this problem is solved solving problems involving nonlinear forms. Therefore,
in this sequence: we will first show the kinematic method of their formation,
- the line up of intersection of l planes α and β is which is characterized by clarity and implements the
constructed; principle of separation, which is widely used in
- the desired point K of the intersection of the line l and computational mathematics.
the plane γ is constructed. In the Oxy coordinate plane, point A, moving according
In the language of linear algebra this problem is to some law, forms a curve 𝑎𝑎1 (y = f(x)) (fig. 2). In turn, the
reduced to solving a system of three linear equations with curve 𝑎𝑎1 , moving in the space Oxyz by its law, forms
three unknowns: by elementary transformations, the ("sweeps") a two- dimensional surface 𝛼𝛼 2(z = γ (x, y)). The
square matrix of coefficients is reduced to a trapezoidal surface 𝛼𝛼 2, moving in the four- dimensional space Oxyzt
one , which in descriptive geometry corresponds to the in the direction of the axis Ot, forms a 3-surface 𝛼𝛼 3 (t =
transformation of one plane into a projecting one.
φ(x, y, z)), which, in turn, "sweeps" the 4-surface 𝛼𝛼 4 (in
Therefore, a simpler calculation of the determinant of the
fig.2 not shown). This process continues until the (n – 2) -
transformed (trapezoidal) matrix corresponds to a simpler
surface 𝛼𝛼 𝑛𝑛−2 "sweeps" the hypersurface 𝛼𝛼 𝑛𝑛−1 .
Fig. 2. Kinematic method for forming multidimensional surface
Thus, the hypersurface 𝛼𝛼 𝑛𝑛−1 is a one - parameter ∞1 The question arises, how to construct a tangent plane
set of (n − 2) - surfaces (stratified into a bundle of (n − 2) to the hypersurface being constructed?
- surfaces). In turn, the (n − 2) - surface is stratified into a In the course of mathematical analysis, the tangent t to
bundle of (n − 3) - surfaces, etc. As a consequence, the the curve m at its point Μ is called the limit position 𝑀𝑀𝑀𝑀 𝑖𝑖
membership problem is solved by constructing an (n − 1) of the secant ΜΝ, which it occupies when the point N
- dimensional nonlinear flag (see [1], p. 2. 2. 3). 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 𝑀𝑀 = 𝑁𝑁 𝑖𝑖 . This definition also Let's consider an example of constructing a tangent
applies to the touch of curved lines, flat and spatial. Since plane 𝜏𝜏 2 to a surface Φ2 in three-dimensional space. In
two curves can intersect at several points, two, three, or textbooks on descriptive geometry, the construction of
more points can coincide in the limit. Therefore, they talk tangent planes to the simplest surfaces (sphere, cone, etc.)
about two-point, three-point, etc. touches. For example, is given. On the surface at this point A we draw two
two second-order curves can have two-point, three-point, graphically simple lines a and b.
and four-point touches. In the language of mathematical The tangents 𝑡𝑡𝑎𝑎 and 𝑡𝑡𝑏𝑏 define the desired tangent plane
analysis, this means that in the case of a two-point touch, τ∋ A. Since engineering surfaces are complex, the curves
the coordinates of the coinciding points satisfy the a and b take the surface sections as planes parallel to the
equations of both curves, and the first derivatives taken coordinate planes of the projections. In our example, the
from the equations of these curves are equal at this point. tangent plane 𝜏𝜏 2 is structurally defined by two tangents 𝑡𝑡𝑥𝑥1 ,
In the case of a three-point touch, the second derivatives 𝑡𝑡𝑦𝑦1 , drawn to the sections 𝑔𝑔𝑥𝑥1 , 𝑔𝑔𝑦𝑦1 of the given surface Φ2
are additionally equal, and in the case of a four-point (Fig. 3).
touch, the third derivatives are also equal. In engineering Structurally, the method of stratification is used to
practice, it is customary to call two-point, three-point, etc. solve such problems. The application of the bundle idea in
touches, respectively, touches of the first, second, and n-th solving problems involving nonlinear forms is shown in
order of smoothness. Curves made up of arcs that touch [8], in particular, by the example of calculating partial
curves are called outlines. derivatives (p. 6. 2. 2). For example, in engineering
These concepts are applied in the appropriate practice, they are used when constructing the tangent plane
interpretation to the touch of surfaces. As noted above, the τ of the surface Φ at its point A.
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 three-
dimensional space.
In differential geometry, it is shown that the set of
tangents 𝑡𝑡 𝑖𝑖 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 𝑡𝑡 𝑖𝑖 forms a conic surface τ with a vertex at
this point [9].
Since the tangent plane τ is uniquely defined by two
straight lines, the algorithm for constructing it consists of
Fig. 3. Construction of the tangent plane 𝝉𝝉𝟐𝟐 to the surface 𝚽𝚽 𝟐𝟐
the following steps:
- through this point A of the surface Φ, any two of its
We generalize the solution of this problem to the
lines a, b are drawn;
construction of a tangent of the 3 - plane 𝜏𝜏 3 to the 3-surface
- at point A, tangents 𝑡𝑡 1 , 𝑡𝑡 2 are constructed to the
Φ3 in four-dimensional space.
selected lines a, b; intersecting lines 𝑡𝑡 1 , 𝑡𝑡 2 define the plane
In the fig. 4 shows an example of generalization of the
τ, touching the surface Φ at point A.
considered algorithm to the construction of a tangent of the
This algorithm is the basis of an analytical method for
3 - plane 𝜏𝜏 3 to the 3-surface Φ3 in four-dimensional space.
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 𝑥𝑥 = 𝑥𝑥𝐴𝐴 , 𝑦𝑦 = 𝑦𝑦𝐴𝐴 , 𝑧𝑧 = 𝑧𝑧𝐴𝐴 , 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(𝑥𝑥,𝑦𝑦,𝑧𝑧) 𝜕𝜕Φ(𝑥𝑥,𝑦𝑦,𝑧𝑧) 𝜕𝜕Φ(𝑥𝑥,𝑦𝑦,𝑧𝑧)
, ,
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
at point A(𝑥𝑥𝐴𝐴 , 𝑦𝑦𝐴𝐴 , 𝑧𝑧𝐴𝐴 ) are the angular coefficients of the
tangents 𝑡𝑡1 , 𝑡𝑡2 , 𝑡𝑡3 , held at point A for curves a, b, c.
The equation of the tangent plane τ has the form:
𝜕𝜕Φ 𝜕𝜕Φ 𝜕𝜕Φ
(𝑥𝑥 − 𝑥𝑥𝐴𝐴 ) + (𝑦𝑦 − 𝑦𝑦𝐴𝐴 ) + (𝑧𝑧 − 𝑧𝑧𝐴𝐴 ) = 0
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
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 Fig. 4. Scheme for constructing a tangent of the 3-plane τ^3 to
variables. the 3-surface F3 in four-dimensional space
Let 3-the surface of Φ3 be given explicitly by the scientific and methodological Council is to organize an
equation: effective system of professional development of teachers.
u = f (x, y, z).
Using the three known coordinates 𝑥𝑥𝐴𝐴 , 𝑦𝑦𝐴𝐴 , 𝑧𝑧𝐴𝐴 of a References
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𝜕𝜕𝑢𝑢 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
curves, the essence of partial derivatives , , . Three universities: traditions and innovations". Perm, 2019.
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
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The real implementation of departments engineering About the authors
graphics of the concept of geometric support of algorithms Dmitrieva Ilzina M., Ph. D. in Pedagogy, Associate
for solving problems of higher mathematics is possible in Professor, Mytischi Branch of Bauman Moscow State Technical
the educational process only in the presence of highly University. E-mail: ilzina@yandex.ru
qualified teachers who possess constructive (graphical) Ivanov Gennady S., Doctor of Engineering, Professor,
and analytical methods for solving geometric problems. Bauman Moscow State Technical University. E-mail:
Unfortunately, several generations of teachers of the ivanov_gs@rambler.ru
Mineev Alexey B., Senior Lecturer, Bauman Moscow State
Department of engineering graphics have considered and
Technical University. E-mail: mineev30@yandex.ru
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