Various
domestic and foreign researchers are engaged in the development of computer
representation of the area of the function of the implicit form
.
Such works include studies [1,2],
which study the application of R-functional modeling in the construction of
grid surfaces for FEM, works [3,4] investigate the method of voxel
representation of vector field components for a three-dimensional function in
mathematical modeling problems, and many others who worked mainly on the task
of rasterization of the function domain for the purpose of zero-boundary
allocation and a positive zone of values for use in various problems of
geometric layout [5-8].
The
method of functional voxel modeling (FVM) was carried out in the system of
“RANOK” [9] and is based on the construction of basic graphical M-images that
display the characteristics obtained as a result of linear approximation of the
surface of the function [10]. By itself, a set of basic images is a graphical
display of local geometric characteristics at each point on a given area for
some algebraic function.
An
image that displays some characteristics of an object is further proposed to be
called an image-model or an M-image. For example, to obtain a set of
two-dimensional base images, the region of the function z = f (x, y) is
linearly approximated, where each point in the given region corresponds to the
equation of the plane
. Increasing
the dimension of the normal to the site, which characterizes the vicinity of the
point by one dimension, we obtain the equation
. As a
result, we have four coefficients (
) for
each argument. After normalization by the norm
we
obtain the components of the normal as local geometric characteristics
for a
local function at each of the points of its domain
:
.
The
voxel representation of the FV-model assumes mapping each local characteristic
into its own image with a monochrome color palette P (for example, in the range
[0, 255]). Figure 1 shows basic M-images
of
local characteristics
, obtained by the formula
An
example was a trigonometric function of the form
, which
is displayed by four M-images (Fig. 1). The question arises: is such a computer
representation of a function applicable in various calculations and
transformations by analogy with an algebraic prototype function?
Figure
1:
An example of a graphical representation of a
functional-voxel model in the area of a trigonometric function
For
further demonstration of arithmetic operations on FV-models, we add to the
consideration an exponential function
of
the form (Fig. 2):
Figure
2:
An example of a graphical representation of a functional-voxel model on the
domain of a trigonometric function
For
further presentation, let us define the basic constructions of the method,
which consist of certain operators. The
FV-method has the following
operators to provide a transition between the analytical and graphical states
of the model:
•
Operator
G:
– operator
of approximation of the function
by the
linear function
, where
- s
the number of voxels in the voxel space of the model. The operator G includes a
partition of the function
on a
given domain into finite elements that are the minimum neighborhood of a point
in this space. In this case, the number of nodes of the finite element
coincides with the dimension of the space. For example, for the function space
the
final element will be a triangle. In general, the operator
G
is
calculated through the determinant:
,
where
–
i-th coordinate of the
j-th node of the finite element.
to
reduce the function
to the
local form, the coefficients
are
divided by the norm
.
•
Operator
– operator
of transition from normal field components to graphic representation of voxel
space. The general view of the operator can be represented as
where
– palette
color intensity resolution, and
– function
space axis number.
•
Operator
– is
the inverse operator of
C
and is calculated by the formula
•
Operator
– operator
defining the value of a function along the selected axis of space
where
the
value of the function along the selected
i-th axis of space, provided
. It
should be noted that checking the condition allows you to assign some minimum
value to
for
example
There are three
possible approaches to solving the problem, leading to a similar result:
•
functional
approach
(F)
- when
the analytical representation of the function space is involved in the
calculations;
•
functional-voxel
approach
(FV)
-
when
voxel data characterizing local geometric characteristics are used in the
construction of local functions for further participation in the calculation;
•
voxel
approach
(V)
- when
only voxel data is used for sequential recalculation of local geometric
characteristics of the model.
Sum.
The
space of the sum of two functions with the same specified area can be written
as a function
. In
this case, each of the functions must be presented explicitly:
,
then
the value of the sum will be written as
. Consider
three approaches to solving the problem using the functional voxel model.
The
functional
approach
(F)
according to the procedure is comparable to the
construction of a new functional-voxel model based on some analytical
expression for the sum of functions
f + g
with the subsequent use of the
operators
{G, M}
to construct
FV-
models.
To obtain a
graphical representation on a given area for the sum function
(f + g)(x)
using the functional approach
(F),
we write the resulting function as:
Using the
operators
G
and
M, we obtain the desired graphical
representation, which will serve as a template for evaluating the remaining
approaches.
Figure
3 shows the basic M-images of the sum of values area
.
Figure
3:
Graphic representation on a given area of the sum of two functions, obtained by
the functional approach (F)
The
functional-voxel approach
(FV)
allows you to abandon the calculation of a complex function of the
type
(F),
using the voxel data of the
FV-
models (Fig. 1 and Fig.
2) for a given range of functions
f
and
g,
respectively.
Through the operator
we
obtain a solution in local functions:
Further
definition of the neighborhood from the obtained points (
allows
us to successively apply the operators
G
and
M
to obtain the
desired graphical representation using the functional-voxel approach.
Figure 4 shows an example obtained
by the FV approach for comparison with the M-images of Figure 3.
Figure
4:
Graphic
representation of the space of the sum of two functions, obtained by the
functional-voxel
approach
Before moving on
to the third approach, consider a variant that combines two types of function
representation in one expression:
Obviously, the
result of the addition to obtain the values of
will
be the same as in the cases considered above. This suggests that the proposed
computer representation of a function can actively participate in complex
analytical calculations on a computer for a given area.
The
voxel
approach
(V)
requires
direct voxel transformations, completely refusing to use the operators
G
and
C
in the
calculation algorithm. This approach does not require the determination of the
neighborhood points and uses only the ratio of the components
and,
if necessary, the calculation of the value of the local function addend at the
point in question on the area.
To
describe approach (V), let us consider in more detail the method of
obtaining the coefficients of the equation of a plane oriented along three
neighboring points:
, where
Consider the
addition of two neighborhoods at a point expressed by local functions:
Let's rearrange
the right side and get
The
operator
G
equally splits the domain into neighborhoods for both
functions
f
and
g. The value of coordinates
for
points for a neighborhood in the plane
coincides
for both terms of the functions.
Hence, it can be argued that
(or
for
the general case). Thus, the coefficients
and
in
the denominator of the expression can be replaced by the general coefficient
which
does not depend on the coordinate
, which
determines the position of the points of the neighborhood relative to the
expressed axis
.
This means that the expression
simplifies form:
,
,
and
.
In the
normalized state
, since
the normalization procedure involves all the coefficients of the corresponding
function equation in the calculation, and they are different for each function.
Therefore,
rearranging the normalized expression leads to
Hence, we can
conclude that now
and
This leads to the
renormalization of the coefficients
Figure 5 shows the
M-images of the graphical representation for the sum of the functions
f
and
g, obtained by the
voxel
approach.
Figure 5:
Graphical representation of the sum of two functions obtained by the
voxel
approach
In the general
case, we have
,
, where
Subtraction.
The subtraction
can be represented as a sum with the negation of a function, which means that
it is enough to consider ways to obtain a negation construction
. Figure
6 shows the voxel model for the function
.
Considering the
functional-voxel
representation, we obtain
Figure 6:
Graphical representation of the negation of the function
, obtained by the
functional
approach
Figure
6 clearly shows that the M-images
,
and
in
relation to the images
,
and
have
an inverted (inverse) color
(where
P
is the maximum value of the palette).
The M-image
in
relation to the M-image
retains
its color values, which is confirmed by the expression obtained above.
Exponentiation.
The
operation of raising to a power can be considered on the procedure of squaring
with further generalization.
The
square power of
f
can be written as
or
Figure
8 shows the
FV-model for calculating such a function using the
functional approach.
Figure
8:
Graphical representation of a function
We will
proceed from the FV-representation of the function
by local functions:
Let's simplify the
expression
from
here
This leads to the
renormalization of the coefficients
Figure
9 shows images of a graphical representation for a given area of the function
f,
squared by the
voxel
approach.
Figure 9:
Graphical representation of a function in a square, obtained by a
voxel
approach
Consider
the reverse action - root extraction for the resulting FV-model
. For
this, we write down that
or
From
here
This leads to the
renormalization of the coefficients
Figure
10 shows the M-images of the graphical representation of a given area for the
function
, obtained
by the voxel approach.
Figure 10:
Graphical representation of the result of extracting the square root
of the function
, obtained by the
voxel
approach
For
comparison, Figure 11 shows an example of a FV-model obtained from the
calculation of the function
Figure 11:
Graphical representation of the extraction of the square root of the
function
, obtained by the
functional
approach
The
proposed result can also refer to the result of
taking modulo.
Multiplication.
Similarly,
consider the arithmetic procedure for multiplying functions by three possible
approaches. As an example, let's keep the same functions:
- trigonometric
and
- exponential.
Then the multiplication function for the first approach takes the form:
Using
the operators
G
and
M, we obtain the required graphical
representation (Fig. 12).
Figure 12:
Graphic representation of the product of two
functions, obtained by the
functional
approach
The functional-voxel
approach (FV),
as noted earlier, allows you to abandon the
calculation of a complex function of the previous type, using voxel data of the
graphical representation of the functions
f
and
g,
respectively.
Through the operator
we obtain a
solution in a normal equation of the form for the operator
:
Further
definition of the neighborhood of the point
allows
us to successively apply the operators
G
and
M
to obtain the
desired graphical representation by the
functional-voxel approach.
Figure 13 shows a set of M-images
obtained by the
FV-approach
for the domain of the product of values.
Figure 13:
Graphic representation of the multiplication of functions obtained by
the
functional-voxel
approach
The
voxel approach
(V)
to
multiplication leads to the determination of normalized coefficients based on
the equation
Let's
change the calculation strategy and write that
and
Both
cases should lead to a single solution, which means
Let's open the
brackets and rearrange the resulting expression:
Hence,
we can conclude that
Obtaining the
components leads to the renormalization of the coefficients
Figure
14 shows the M-images of the graphical representation for the product of the
functions f and g, obtained by the
voxel
approach,
where you can see a complete match
with Figure 13.
Figure
14:
Graphic representation on a given area of the product of two functions,
obtained by the
voxel
approach.
In the
general case, we have
Divide.
Consider
the arithmetic procedure for dividing functions in three possible approaches. The
functional dividing for the first approach will take the form:
Using the
operators
G
and
M, we obtain desired graphical representation
shown in Figure 15.
Figure 15:
Graphical representation on a given area of the quotient of two
functions, obtained by the
functional
approach
Functional
voxel approach (FV).
Through the operator
we obtain a
solution in a normal equation of the form for the operator
:
Further
definition of the neighborhood of the point
allows
us to successively apply the operators
G
and
M
to obtain the
desired graphical representation by the
FV
-approach. Figure 16
shows the M-images obtained by the
FV-approach,
where you can see a
complete match with Figure 15.
Figure 16:
Graphical representation on a given area of the quotient of two
functions, obtained by the
functional-voxel
approach
The
voxel
approach
(V)
of
functional dividing leads to the determination of normalized coefficients based
on the equation just considered.
We
proceed from the fact that the quotient can be represented by the product:
Having
previously obtained the
FV-model for the
function,
you can use the product procedure that was just discussed above.
Figure
17 shows the M-images of the graphical representation for the private function
, obtained
by the functional approach (F).
Figure 17:
Graphical representation on a given area for the
function, obtained by the
functional
approach
Figure
18 shows the M-images obtained as a result of the voxel approach to the product
A visual assessment of the result
obtained with the results in Figures 16 and 17 allows us to speak about the
correctness of the formulas obtained for calculating the
V-divide.
Figure 18:
Graphical representation on a given area for the
function, obtained by the
voxel
approach
The
proposed set of approaches for the implementation of means of constructing and
calculating analytically presented implicit functions on a given domain allows
us to create a new computer computing platform that makes it possible, based on
simulated local geometric characteristics, to significantly simplify the
representation of complex functions domain to its local analogues (local
functions) on a computer and, based on them, to organize complex computational
constructions in the form of functional expressions. An example is the
FV-modeling of R-functions that allow to obtain set-theoretic operations on the
domain of two analytical expression [11, 12], etc. Also, the implementation of
basic arithmetic procedures allows us to talk about the possibility of using
voxel models in solving functional equations, avoiding complex formulations.
[1]
Lysniak
A.A, Choporov S.V., Gomeniuk S.I. Metodika vizualizacyi geometricheskih obyektov,
opisannyh s pomoshyu R-funkcyi // Vestnik Zaporozhskogo nacionalynogo universyteta
: Sbornik nauchnyh statey. Fiziko-matematicheskie nauki. – 2010. – ¹1. – S.
88-97.
[2]
Choporov
S.
V.,
Lysniak
A.
A,,
Gomeniuk
S.
I.
Ispolzovanie
funkcyi
V.
L.
Rvacheva
dlya
geometricheskogo
modelirovaniya
oblastey
slozhnoy
formy
//
Prikladnaya
informatika. – 2010. – ¹2
(26). –
S. 109-122.
[3]
Myltsev
O.
M.,
Kondratyeva
N.
O.,
Leontyeva
V.
V.
Funkcionalnaya
model
osnovnyh
biznes
-
processov
systemy
“RANOK” //
Vestnik
Zaporozhskogo
nacionalynogo
universyteta:
Sbornik
nauchnyh
statey.
Fiziko-matematicheskie nauki. – 2018. – ¹2. –
S. 88-99.
[4]
Myltsev
O., Pozhuyev A., Leontieva V., Kondratieva N. The Assessment of the Complexity
of the Recursive Approach to Voxelization of Functionally Defined Objects in
the Euclidean Space En.
International Journal Of Mathematics And Computer
Research. 2020. Volume 8. Issue 03. P. 2028-2034. (Index Copernicus).
[5]
E
Vin Tun, Markin L.V., Turbin N.Yu. Geometricheskiy podhod k ocenke
ergonomichnosti proektiruemyh izdeliy // Izvestya TulGU. Tehnicheskie nauki. –
2021. – Vyp. 4. – S. 469-478.
[6]
Markin
L.V., Korn G.V., Kui Min Han, [16] E Vin Tun Diskretnye modely
geometrycheskogo modelirovanya komponovki aviacionnoy tehniki // Trudy MAI. –
2021. – Vyp. 86. – S. 1-30.
[7]
Akenine-Moller,
T. Real-Time Rendering. Third Edition / T. Akenine-Moller, E. Haines, N.
Hoffman. – Wellesley: A. K. Peters, Ltd., 2008. – 1027 p.
[8]
Crassin,
C. GigaVoxels: Ray-Guided Streaming for Efficient and Detailed Voxel Rendering
/ C. Crassin, F. Neyret, S. Lefebvre, E. Eisemann // ACM SIGGRAPH Symposium on
Interactive 3D Graphics and Games (I3D), feb 2009.
[9]
Maksimenko-Sheyko
K.V., Sheyko T.I., Tolok A.V. R-funkcii v analiticheskom proektirovanii s
primeneniem sistemy “RANOK” / Vestnik MGTU Stankin. M.: MGTU “Stankin”, 2010. ¹
4. S. 139-151.
[10]
Tolok
A.V. Lokalnaya komputernaya geometria : uchebnoe posobie / A.V. Tolok. – Moskva: Ay Pi Ar Medya, 2022. – 152 s.
[11]
Tolok
A.V., Tolok N.B., Sycheva A.A. Construction of the Functional Voxel Model for a
Spline Curve / Proceedings of the 30th International Conference on Computer
Graphics and Machine Vision (GraphiCon 2020, St.Petersburg). Sankt-Peterburg:
CEUR Workshop Proceeding, 2020. Vol-2744. Ñ.
http://ceur-ws.org/Vol-2744/paper52.pdf.
[12]
Maksimenko-Sheyko
K.V. R-funkcii v matematicheskom modelirovanii geometricheskih objektov I
fizicheskih poley : monographia / K.V. Maksimenko-Sheyko. – Harkov : Izd-vo IPMash
NAN Ukrainy, 2009. – 306 s.
