Most of the information about the
world around a person is received through sight. In the process of visual
perception of an object, researchers, above all, pay attention to its shape and
color. The majority of computer-aided design and design systems are built on a
similar property of human perception of information. This approach allows us to
convey not only information about the shape of the test (or projected) object,
but also additional properties as color support, reflecting the change in the
physical, mechanical, architectural and other properties of the object.
However, such a color reproduction of in-formation is either a discrete set of
colored zones (e.g., computer-aided design systems, the finite element analysis
[1-2]), or a linear relationship (e.g., geo-information systems with or without
using a gradient [3] it [4]).
In this paper, it is proposed to
use continuous nonlinear color coding of information to expand the capabilities
of visualizing multidimensional space, in which the color change is a certain
continuous function.
For this, it is necessary that the
analytical description of the multifactor process (or phenomenon) under study
be presented in parametric form, in which each individual axis of the global
coordinate system corresponds to only one specific characteristic of the simulated
geometric object. A similar approach is used to visualize multidimensional
geometric objects by projecting them on the projection plane [5]. Only in this
case, projection does not occur on the plane, but on the axis of the
projections.
The BN-calculus (the
Balyuba-Naydysh calculus [6-8]) was developed in the 90s of the 20th century by
a team of scientists from the Melitopol School of Applied Geometry. By origin,
BN-calculus is a synthesis of vector, tensor, and barycentric calculi, from
which it borrowed ideas and methods for determining geometric objects at the
stage of formation, and their analytical description [6]. Initially, the
BN-calculus was created as a special mathematical apparatus for engineering
calculations related to modeling curves and surfaces of any shape in accordance
with predetermined requirements and in the required parameterization. However,
in the process of its development, in addition to shaping geometric objects, it
found wide application in the field of modeling and optimization of multifactor
processes and phenomena using multidimensional interpolation and approximation
in their geometric interpretation [9-11].
The basic element of the BN-calculus
is a point, and all geometric objects are defined as an organized set of
points, for the analytical description of which the invariant properties of the
parameter with respect to parallel projection are used.
Proceeding from this, the BN
calculus in the affine space can be considered a special case of the Wurf
calculus in the projective space proposed by H. Staudt.
The main distinguishing feature of
the BN calculus is that each projection of the desired geometric object on the
axis of the global Cartesian coordinate system is determined separately. And
the geometric object itself is the result of the joint interaction of all
projections.
Thus, using the color scheme, it becomes
possible, along with the geometric shape, to convey some additional properties
that correspond to one of the axes of the global coordinate system.
It should be noted that the
selection of one of the parameters using color coding is a fairly common
technique for constructing a visual representation, which is an effective tool
for scientific visualization for systematization, modeling and analysis of multidimensional
data [12-13].
The point equations inherently represent
a specific symbolic record, which includes a collection of simplex points and
some functions of the current and fixed parameters. For the practical use of
point equations, it is necessary to determine the starting points in a certain
coordinate system. Such a procedure in the BN-calculus was called the coordinate-wise
calculation, the result of which is a system of the same type of parametric
equations. Moreover, each of the parametric equations is geometrically a projection
of the desired geometric object onto one of the axes of the global coordinate
system. Coordinate-wise calculation is one of the key elements of the theory of
BN calculus and its essence is easiest to consider with a simple example.
Let two points be given:
,
. The equation of the
line has the form:
. It is easy to notice the
remarkable peculiarity of this formula - the identity of the expressions
between the signs of equality, regardless of the dimension of space. If the
constant ratio of the difference in the coordinates of the points is taken as a
parameter
:
,
and then enter a convenient character
notation for this coordinate expression system:
, then
we get a compact notation of the equation of a straight line in point form. We note
that the system of coordinate-wise equations allows one to perform algebraic operations
in point form:
Moreover, all arithmetic operations
are performed with points as with ordinary numbers.
The last point equation defines a
point
at the value
, point
– at
, line
segment
– at
. We conclude
that the point equation of the segment is provided by a special selection of
the parameter. Obtaining required points using point equations determines the
mathematical apparatus of BN calculus. In this case, the parameter should
ensure the identity of mathematical operations with points, i.e. with their coordinates.
Since the graphic image of a point in a Cartesian coordinate system is based on
parallel projection, the invariant of such a projection should be a parameter -
a simple ratio of three points of a line.
For example, moving from a point
equation of a straight line to a system of parametric equations, for
three-dimensional space we get:
Moreover, the number of parametric
equations directly depends on the dimension of the space in which the desired geometric
object is located.
Similarly, replacing the points
with the coordinates corresponding to them, a transition is made from symbolic
point equations to a system of parametric equations.
The use of a system of parametric
equations for visualizing geometric objects is more preferable since it allows not
only to unambiguously determine the type of geometric object by a parameterized
attribute, but also to effectively use separate equations corresponding to
projections on the axis of the global coordinate system for coding of various
properties of the simulated object.
In geometric modeling of
multifactor processes and phenomena [9–11], a curved line, as a one-parameter
set, serves as an analytical description of one-factor processes and phenomena.
It should be noted that a straight line is a special case of a curve of a line
with zero curvature. Moreover, for the visual representation of a straight
line, it is sufficient to have one-dimensional space just by this line both formed
and limited to a certain segment. For graphical visualization of a curved line,
it is necessary to use at least two-dimensional space — a plane, in case of a
flat curve (visualization of spatial curves is discussed below). However, if
you use color coding of a curve line, you can limit yourself to one-dimensional
space. Then the problem is reduced to representing a straight line on which the
color changes with a change in curvature. The easiest way to organize such a
visualization is to use separate projections of a flat curve of a line on the
axis of a flat Cartesian coordinate system, the analytical reflection of which
is a system of parametric equations.
Let us consider an example of
visualization of a plane parabola passing through 3 predetermined points [14].
The point equation of a parabola has the following form:
where
‒
current point of parabola arc;
‒
points through which the parabola arc passes;
− current
parameter that varies from 0 to 1;
‒
padding the parameter to 1.
Passing from the point equation to
the parametric, we obtain:
The visualization of such a curve
in the plane will have the following form (Fig. 1).
Fig.
1. Visualization of a flat parabola
Next, we construct a line segment
, whose color will change depending on the position
of the current point. So, the movement of the current point will be determined
by the equation
, and the color of the segment for
each position of the current point −
. As a
result, we obtain the following visualization of the parabola arc (Fig. 2), for
which in order to restore the values
of the
function
you can use the attached color legend in
the form of a spectrogram.
Fig. 2.
Visualization of a parabola in which the coordinate change is color coded
It should be noted that the
obtained color segment of the straight line is a projection of the parabola
arc, the maximum of
, which is achieved at a value
that corresponds to the violet color. Moreover, Figure 2 clearly shows how the
curve first increases to the maximum point, and then decreases.
Thus, the use of color coding
allowed us to reduce the dimensionality of the space of the geometric object by
one during the visualization process.
Modeling of spatial lines which
have more degrees of freedom compared to flat lines has its own distinctive features.
The term "spatial curve" is usually understood as a line of double
curvature, the points of which do not lie in the same plane. In a sense, this
definition is contradictory and holds true exclusively for 3-dimensional space.
For example, for a curve of the 4th order, which generally belongs to 4-dimensional
space, the points also will not lie in the same plane, moreover, they will not
lie in the same 3-dimensional space.
The same is true for algebraic
curves of
order. However, to fully visualize the shape
of spatial curves, you can only imagine space of dimension
, which is an extremely difficult task.
Therefore, to reduce the dimension of space, the projection method is used,
most often orthogonal. Thus, any spatial curve can be represented as a set of
projections on the projection plane of the global coordinate system, which is
not very convenient for analytical calculations, for which projection is not
preferred on the projection plane, but on the axis of the global coordinate system
.
When considering algebraic curves of
the
order, it should also be taken into account
that they are a collection of
sets existing in spaces
of various dimensions. For example, a third-order curve can be represented in
the form of two sets [15]: a flat curve (cube) and a spatial curve (belonging
to a 3-dimensional space).
Now the visualization of spatial
lines is carried out by means of computer graphics, mainly by approximation.
For example, the work [16] gives an example of the capabilities of the 3dMAX
visual modeling program that allows you to visualize a spatial curve by skipping
circular sections along it.
Better visualization results can be
obtained using color coding of one of the coordinates of the system of parametric
equations of the spatial curve. At the same time, computer mathematics systems
are used to visualize spatial curves. This approach allows us to increase the
dimension of the space for visualizing algebraic curves of higher orders.
Consider an example of
visualization of a third-order spatial Bezier curve in the form of a flat
projection onto the horizontal plane of projections, which is determined by the
following point equation:
where
‒
3rd arc Bezier curve current point;
‒
points determining the position of the tangents
and
3rd
order Bezier arcs.
Having performed the
coordinate-wise calculation of the point equation of the arc of a third-order
Bezier curve in a 3-dimensional space, we obtain:
Fig. 3.
Visualization of the arc of the curve of the 3rd order in 3-dimensional space
In fig. 3 spatial arc of a curve of
the 3rd order is shown from different angles. Such a need arises, since in a
static image from one angle it is not obvious that the arc of the curve is a
line of double curvature, and not a flat line.
We use a combination of functional
dependencies of the coordinates
and
of the current point
to determine the position of the horizontal
projection of the spatial curve, and set the elevation (
point’s
coordinate) by changing the color palette.
The results of visualization of the spatial Bezier curve of the 3rd order are
presented in Fig. 4.
Fig. 4.
Visualization of the spatial curve of double curvature on the plane
Color coding can also be used to enhance
the effect, to emphasize some properties of the simulated object. For example, Figure
5 shows the visualization of a spatial curve for which the elevation values
duplicate the change in the color of the curve arc along the current
point.
Fig. 5.
Visualization of a curve of double curvature in 3-dimensional space
The proposed approach to the visualization
of spatial lines can be generalized and used to visualize spatial curves belonging
to 4-dimensional space. Of algebraic curves, curves starting from the fourth
order possess the necessary properties.
Consider an example of
visualization of an arc of a fourth-order curve passing through the 5 ahead
given points [17] in 4-dimensional space:
where
The system of parametric equations
of the arc of a 4th-order curve in 4-dimensional space will have the following
form:
In accordance with the proposed visualization
method, we use the first three equations from the system of parametric equations
to determine the position of the current point in 3-dimensional space, and we
can represent the fourth equation by changing the values
of the color palette. As a result, we obtain a visualization of one
of the projections of the spatial arc of a 4th-order curve belonging to
4-dimensional space (Fig. 6).
Fig.
6. Visualization of the projection of the spatial arc of a 4th-order curve on
3-dimensional space
Of course, not all spatial lines
are conveniently visualized in this way. For example, visualization of a
helical cylindrical line will not be possible, since the projections of the current
point of the curve will overlap each other. In many cases, the proposed
visualization method can serve as an effective tool for acquiring new knowledge
about the geometric properties of objects in 4-dimensional space.
Visualization of the surface is not
a trivial task in itself, since it requires the placement of a
three-dimensional image in the plane of the screen. Axonometric and perspective
projections, discretization, color change, and other visualization technologies
are used here. Moreover, the surface itself can be represented as a set of
discrete points, lines, triangles, rectangles and other elements. A separate
type of visualization, including surfaces, are nomograms, which are widely used
in engineering practice and represent a discrete set of lines with certain
properties that usually correspond to fixed values
of one of the factors under study. In this case, nonlinear grids
(for example, logarithmic) are often used.
One of the ways to visualize such
nomograms can be by using color coding of one of the coordinates, which also
allows to reduce the dimensionality of the space for analyzing the results.
As an example, let us consider a
visualization of a two-factor process of the dependence of heat
received from a boiler unit on the
temperature head of a heat carrier
and the diameter of
convective pipes
(Fig. 7).
Fig. 7.
Visualization of a two-factor process in the form of a nomogram
Another, more effective way of
visualizing the surface compartment using color coding is to use not the
discrete lines shown on the nomogram, but a continuous color field of values
(Fig. 8).
Fig. 8.
Visualization of a plot of a topographic surface using color coding of a
coordinate change
Figure 8 shows an example of
visualization of a section of a topographic surface, the geometric model of
which is presented in [18], using color coding of changes in elevations
corresponding to changes in coordinates
.
A similar visualization method is
widespread in the two-dimensional problems of computational physics to
represent the studied quantity as a function of two spatial variables. For
example, in [19-20].
The paper proposes an approach that
allows visualization of geometric objects based on the point equations of their
formation. This approach allows for visualization to reduce the dimension of
the space, which, in future, makes it possible to visualize more properties
that characterize the projected or investigated object. The given examples
confirm the possibility of using the proposed approach in engineering and
scientific practice for visualization.
Of course, like any projection apparatus,
color coding of coordinate changes has its drawbacks. The main one is the
complexity of representing a geometric object in space, which requires the user
to have an advanced spatial thinking skill. Nevertheless, it can be effectively
used as one of the alternative tools for engineering and scientific
visualization, which has its own advantages.
The prospect of further research is
visualization using the point equations of geometric bodies and hypersurfaces
as a multi-parameter set of points belonging to a multi-dimensional space.
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