The purpose of the
conducted research was to study the possibility and methods of application of
isotropy
of a local function
which is the one of the important properties in local
computer geometry, underlying the fundamental difference of this section of
geometry from the section of differential geometry [1-5]. Such approach significantly
expands the range of applied problems solved by local geometric modeling. The basic
means of representing information in local computer geometry [6, 7] are local
geometric characteristics for the neighborhood of points on a given region with
dimension
,
describing a
homogeneous unit vector
.
The components of
such a homogeneous unit vector
determine the local
function
at each point of
the region
.
The local
function, in turn, describes a linear law duplicating at a given point any
other law specified by the analytical representation
(Fig. 1).
Fig.1. Duplication of the law
F(x)=0
by a local function at a point
It is known that the
derivative of the function gives the slope of the tangent line to the curve at
this specific point, which means that it can be applied as a linear law at this
point, replacing the law given by the original function. This property leads to
a simplification of analytical calculations at this point, which allowed Isaac
Newton and Gottfried Wilhelm Leibniz to develop the theory of differential and
integral calculus. In fact, it is easy to show that tangential differential
calculus can be attributed to a special case of local geometric calculus in
general, and especially in the considered case of local computer geometry,
since only modern computer technologies have made it possible to process large
amounts of data for the development of such calculus.
For simplicity and clarity, let us consider the
principle of isotropy for the two-dimensional case. Let's turn to Figure 2 and assume
that the local function passes through a point orthogonally to the tangential
direction.
Fig.2. Duplication of the law
by an orthogonal
local function at a point
It can be argued that such a local function also
duplicates the law of the function
(or explicitly
)
as well as the
tangential local function, since the following relation is preserved:
|
|
(1)
|
It is interesting that for any local functions
describing the equation of a straight line passing through a selected point,
the property of duplicating the function
is preserved. We can
say that a "bundle" of local functions at a single point can
duplicate any function
passing
through this point. That is, for a local function at the point, the principle
of isotropy is observed:
the uniformity of ratios
of the local function
arguments
in all orientations.
Using the example of a function of three variables
,
we will show that for any
arbitrarily given components
of a homogeneous
vector of a local function, it is always possisble to calculate the fourth
component
,
leading to an
unambiguous determination of the range of the original function
.
Let's consider an example of describing the range of
function values for the zero contour "square". We will
describe such a range using R-functional modeling tools:
|
|
(2)
|
Function (2) provides a zero value along the contour of
a square with a side equal to two units and centered at the origin. For
consideration, we will select a 4x4 region centered at the origin.
In order to obtain the domain
of tangential local functions, i.e. the domain of tangent planes, it is
sufficient to create a regular grid covering for the given domain. To define a
triangular element of the plane, for each vertex we add two neighboring vertices
shifted along the
and
axes by the grid step. For
each of these three vertices we calculate
z
using formula (2) and
formulate a local function using the linear operator [6,7]:
|
|
(3)
|
Reducing the components to a uniform unit vector and
matching the monochrome color palette gradation
on the
M-image, we
obtain:
|
|
(4)
|
Figure 3 shows M-images that display on a computer the range
of all four characteristics
,
respectively.
Fig.3.
M-images
describing the range of tangential local functions for a zero square contour
Figure 4 shows the positive range of values of the
function (2) in monochrome, and the blue colour represents negative values of
this function.
Fig.4.
Positive and negative value ranges of the function (2) superimposed on the
image
To begin with, let's model a situation where the local
function is constant and horizontal at each point, i.e.
.
At the same time,
the first two images -
and
- possess a gray
color corresponding to the zero value, and the image
takes on a white
color corresponding to a unit value. Now we have to to determine the fourth
image
,
which carries
basic information about the function (2). To do this, at the current point
i
with coordinates
,
calculate
using formula (2)
and find the coefficient
:
|
|
(5)
|
We will reduce the obtained components
to the components
of a homogeneous unit vector
,
and then to the
correspondence with the monochrome color palette P according to formula (4), we
will obtain the
M-images shown in Figure 5.
Fig.5.
M-images describing
the range of local functions for the zero square contour generated by the last
component
- for
The first two
M-images keep the color constancy
equal to the value 127 (gray), obtained after normalization, for the zero value
of the cosine of the angle deviation with a reference to the
and
axes. During normalization, the
third
M-image
is influenced by
the values of the norm, since the numerator in this case is not zero as in the
first two cases.
Let us consider the value of
at each point of a given
region using a local function according to the formula
|
|
(6)
|
We will display the range of negative
values of
in
blue, making sure that it matches Figure 4 (Fig. 6).
Fig.
6. Image of the positive and negative range of
values of local
functions
The orientation of the planes described by local
functions that are orthogonal to the
and
axes, respectively, is shown
in Figures 7 and 8. Here we can observe how the component
influences the
color of the
M-image of the corresponding component
,
equal to one.
Fig.7.
M-images
describing the range of local functions for the zero square contour generated
by the last component
- for
.
Fig.8.
M-images
describing the range of local functions for the zero square contour generated
by the last component
- for
.
The conducted studies have
shown that the developed approach allows using a single image
,
with known
components
expressed by
constants, to obtain the domain of local functions describing the domain of
function (2) by means of the proposed key encryption algorithm. The idea is to
use the procedure of replacing the entire image containing the constant color
at each point with one numerical value. For example, in the special case considered
in Figure 5, the M-images
,
and
can be represented
by the numbers 0,0,1, and cases represented by Figures 7 and 8 contain the
numerical combinations 0,1, 0 and 1,0,0 in M-images. Thus, for any of the
selected combinations with the available values of
,
and
,
it is sufficient
to calculate the value of
at each point of
the corresponding image
.
In this case, the
combination of
,
and
is the key to
make the choice between three considered images
.
Let's consider the
algorithm for calculating
.
We have already determined that
the value of
at the point of the
image
is calculated by
the formula
|
|
(7)
|
With the values
of components
,
coefficient
can be expressed
from formula (4). In the considered case
|
|
(8)
|
As a result, we obtain:
|
|
(9)
|
It follows that each of the
three M-images
in Figures
5, 7, 8, together with its numeric key, contains sufficient initial graphical
information to obtain the same function domain (2). This means that it is
possible to completely restore the Functional Voxel model shown in Figure 3 if
any of the M-images are available together with a numerical key. Note also that
as the dimensionality of the considered space of function arguments increases,
the number of zero components will increase correspondingly, i.e. formula (9)
remains universal, while the index for
and
increases.
Now let's consider the more complicated
problem by assigning different values of integer type to three constants (for
example,
).
Using formula
(5), we obtain the value of
(Fig. 9).
Fig.9.
M-images describing the range of function (2) for
Let's try to express
again, given that
all three components
are known to be some
numeric values other than zero. Then
|
|
(10)
|
that leads to
|
|
(11)
|
where
– a complex key
encoding for an M-image
.
Figure 10 depicts the result of
calculating the negative and positive ranges of
values for local functions
composed by M-images represented in Figure 9.
Fig.10. Positive (gray) and negative (blue) ranges of
for local functions with the
key 2,6,1.
It is not difficult to show that any three images from
this composition of the functional voxel model, replaced by constants, make it
possible to generate one of the
M-images to describe the function
domain.
In [6,7,8],
computational operations on local functions using an M-image representation of
a given domain are considered. A typical example of the isotropy occurs when obtaining
the result of component-by-component multiplication of the values of two
different functions represented by homogeneous vectors at points in a given
area.
As an example, let
us consider different types of functions:
-
trigonometric
f-function (Fig. 11)
|
|
(12)
|
and exponential
-function (Fig. 12)
|
|
(13)
|
Fig. 11.
M-images representing the domain of
local functions for the trigonometric function (12)
Fig.12.
M-images representing the domain of local functions for the exponential function
(13)
The benchmark
for comparing those results will be images of local functions obtained directly
by the multiplication of functions (12) and (13), shown in Figure 13.
Fig.13.
M-images representing the domain of local functions for the product of
functions (12) and (13)
In [6,7,8], the
solution to the problem of determining the product based on local functions of
a homogeneous vector is proposed to be expressed as follows:
If in the previous
description, Figure 13 demonstrates tangential local functions obtained by
linear approximation using formulas (3) and (4), then the result of
calculations using formula (14) is not tangential (Fig. 14), but the
-surface
differs in
the 14th decimal place at each point of the considered domain.
Fig. 14.
M-images
describing the domain of local functions according to formula (14)
It is clear that
there is another solution that allows you to express the components:
In this case, the
M-images
will, of course, show a different picture (Fig. 15), however, the
surface will coincide
with the previously considered cases.
Fig.15. M-images
describing the domain of local functions according to formula (15)
Thus, due to the
existence of the isotropic property, the commutative property of multiplication
is preserved in the arithmetic operations of local geometry.
The question arises:
is it possible to obtain tangential local functions from any
M-image
representation of a function?
Let us consider an
example describing the process of generating the set of
M-images represented
at Figure 3 directly from the image
displayed at Figure
9. To do this, we transform the color value of each current point of the image
into the
value of the fourth component of the homogeneous unit vector
in terms of expression
(6) and express the component
using given conditions
:
|
|
(16)
|
The expression of
color gradation according to formula (4) leads to the restoration of all four
M-images of Figure 9:
|
|
(17)
|
The resulting
M-image
representation provides the
z-value of the function (2) at the corresponding points
of the domain:
|
|
(18)
|
At this stage,
taking into account the discreteness of the
M-images, we should apply a
linear approximation of the function domain by restoring ordered triples of vertices
of a rectangular grid, following the calculations on the basis of formulas (3)
and (4).
The conducted
studies have shown the wide possibilities of computing tools and forms of
representation of a multidimensional domain of a complex function on a computer
when using the components of a homogeneous vector to describe a single point in
space. The revealed isotropic property significantly expands the methods of
such representation and allows to pack the computer data into a single
M-image, reducing the
remaining images to constant values, which helps to solve the problems of
graphical encryption of the object geometry. It is shown that the isotropy
simplifies the automation of algebraic and arithmetic calculations over
functions, which allows the implementation of complex computational structures,
such as R-functions [6], etc., using local functions. This paper also presents
a method for transitioning to the traditional tangential position of local
functions for performing differential and integral calculus [9-11].
The research was
carried out within the framework of the scientific program of the National
Center for Physics and Mathematics, direction No. 9 "Artificial
intelligence and big data in technical, industrial, natural and social
systems".
1. Egorov A.I. Ordinary differential equations with applications. Fizmatlit, Moscow, 2003. 384 pp.
2. Romanko V.K. Differential equations and the calculus of variations. Moscow: Laboratory of Basic Knowledge, 2000. 344 pp.
3. Stepanov V.V. The Course in Differential Equations. M.: Editorial URSS, 2004. 472 pp.
4. Konev V.V. Partial differential equations: Study guide / Tomsk Polytechnic University.: https://portal.tpu.ru/SHARED/k/KONVAL/notes/Partial.pdf.
5. Krasnov M.L., Kiselev A.I., Makarenko G.I. Ordinary differential equations: Problems and examples with detailed solutions: Study guide. Moscow: LENAND, 2019. Р 256 pp. (Higher mathematics in problems and exercises)
6. Tolok A.V. (2022) Local computer geometry. Study guide. IPR Media, ЊoscowР 147 pp.
7. Tolok, A.V. (2016) Functional Voxel Method in Computer Modeling. Fizmatlit, Moscow Р 112 pp.
8. Alexey Tolok, Natalya Tolok. Arithmetic in Functional-Voxel Modeling (2022). Scientific Visualization 14.3: 107 - 121, DOI: 10.26583/sv.14.3.08
9. Tolok A.V., Tolok N.B. Differentiation and Integration in Functional Voxel Modeling // CONTROL SCIENCES. 2022. Ь 5. P. 51-57.
10. Tolok A.V., Tolok N.B. The Functional Voxel Method Applied To Solving a Linear First-Order Partial Differential Equation with Given Initial Conditions // CONTROL SCIENCES. 2023. Ь 6. P. 65-71.
11. A.V. Tolok, N.B. Tolok. Functional-Voxel Modeling of The Cauchy Problem (2024). Scientific Visualization, 2024, v. 16, N 1, p. 105 - 111, DOI: 10.26583/sv.16.1.09
