International Journal of Electrical and Computer Engineering (IJECE) Vol.
No.
June 2014, pp.
ISSN: 2088-8708
On the Analogy of Non-Euclidean Geometry of Human Body With Electrical Networks Alexander Penin*.
Anatoli Sidorenko*.
Ashok Vaseashta** * D.
Ghitu Institute of Electronic Engineering and Nanotechnologies.
Chisinau.
Moldova ** IASC/ICWI.
NUARI, 13873 Park Center Rd.
Ste 500.
Herndon VA.
USA
Article Info
ABSTRACT
Article history:
A review of application of non-Euclidean geometries for interpreting the process of the growth in the human body is presented and features employing non-Euclidean geometries in the electric circuit theory are modeled.
Growth of the human body and changes of parameters of an operating regime of an electronic network correspond to projective and conformal transformations which possess an invariant being the cross-ratio of four points.
The common mathematical apparatus represents interdisciplinary approach in view of analogy of processes of a different physical nature.
The results obtained here demonstrate development of a methodology of application of non-Euclidean geometries and its biological correlation to the growth of human body.
Received Jan 13, 2014 Revised Apr 20, 2014 Accepted May 8, 2014 Keyword:
Electrical network Human body Mybius equivalent Non-Eucldean geometry Copyright A 2014 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Ashok Vaseashta.
IASC/ICWI.
NUARI, 13873 Park Center Rd.
Suite 500 Herndon.
VA USA Email: prof.
vaseashta@ie.
INTRODUCTION
The growth of the human body is essentially a nonlinear process for ordinary or Euclidean Analysis of the structure and growth reveals that separate body parts or all the three-component kinematics blocks .
he phalanxes of fingers, the three-membered extremities and the three-membered bod.
change according to Mybius transformations which are characteristic for conformal and projective geometry.
Such transformations possess an invariant or invariable value.
Therefore, all the above mentioned threecomponent blocks are characterized by a constant value throughout the human life .
, .
From this point of view.
Euclidean geometry appears as a possible analytical instrument.
The advent of the special theory of relativity led to a new term.
Augeometrization of physicsAy, which is a methodological doctrine and may be defined as the application of geometrical methods in physics, whenever possible.
This is formally the theory of invariants of some group of transformations (Poincary-Lorentz grou.
, or space-time geometry.
The basic principles of projective geometry which are reflected in methods of linear perspective.
The geomaterization of physics, biology, and neuroscience covered a multitude of fields and as a result of its biological application, there is an image of a subject in which both distances and corners change.
It is important to note that these changes appear neither arbitrarily or randomly, but are invariant being the cross- ratio of four points on a straight line .
double proportio.
Geometry of the Euclidean space complemented by one infinitely remote point, is called the conformal geometry .
Projective interpretation in the form of a stereographic projection of points of the sphere to the plane gives an example of the conformal plane .
s an example of such stereographic projection is cartograph.
, when the value of the cross- ratio of four points remains constant .
Such plane projection is applied in physics for solving of electrostatic problem.
Journal homepage: http://iaesjournal.
com/online/index.
php/IJECE A ISSN: 2088-8708 Mybius's certain subgroup of transformations corresponds to LobachevskiAos geometry or hyperbolic geometry, the so-called Poincare's conformal interpretation .
It indicates that the space of visual perception is characterized by Lobachevski geometry.
A number of publications show that in electric networks, the changes of operating regime parameters can be interpreted as projective and conformal transformations.
addition, the relationship of regime parameters at different parts of a network also is being described by projective transformations .
, .
, .
An interdisciplinary approach applies common mathematical apparatus in various areas of science and the similarity of processes of different physical nature.
As the example of such interdisciplinary approach, we present a review of application of non-Euclidean geometries for interpretation of the human body growth process, and the features of use of non-Euclidean geometries in the electric circuit theory are shown.
ON THE FEATURES OF GEOMETRIC TRANSFORMATIONS
Projective transformations General case: Generally, the projective transformation of points of one straight line U L into the points of the other line RL is set of points with the projection center S or the three pairs of respective points .
hown in Figure .
The projective transformations preserve the cross ratio of four
RL2 A RL1 RL3 A RL1
m A (R R R R ) A 2
RL A RL4 RL3 A RL4
m A (U L1 U L2 U L3 U L4 ) Figure 1.
Projective transformation of straight lines points Affine transformation: There is the projection center S, but the straight lines are U L .
RL parallel.
Therefore, the invariant of an affine transformation is the simple ratio or proportion of three points.
Euclidean transformation: If the projection S C Cu center and the straight lines U L .
RL are parallel, the projection is carried out by parallel lines.
This projection corresponds to the Euclidean transformation, which is parallel translation of a segment.
The Euclidean transformation preserves the difference of points.
The transformations of an initial rectangular coordinate grid, considered here are presented in Figure 2.
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Stereographic projection: The projection of points of the sphere U i (U 1 .
U 2 ) from the top pole on the tangent plane n1 , n2 placed at the bottom pole is presented in Figure 3.
For the sake of simplicity, the coordinate axes U 1 .
U 2 and n1 , n2 are coincident.
Figure 2.
: Characteristic transformation of the Cartesian grid by various group transformations, .
Euclidean, .
: affine, and .
: projective.
Figure 3.
Stereographic projection of the sphere on the conformal plane n1 , n2 The conformal plane differs from Euclidean by existence of one infinitely remote point corresponding to the top pole of the sphere.
On the Analogy of Non-Euclidean Geometry of Human Body With Electrical Networks (Ashok Vaseasht.
A ISSN: 2088-8708 Conformal transformations: The area of changes of the values U1 ,U 2 corresponds to the sphere equator in Figure 4a.
Let the value U1 to be.
U1 A const .
hat is the line L1 ).
The circular section L2 ison the sphere while the circle L3 ison the plane n1 , n2 .
The similar family of circles is described by the rotation group of sphere, as it is shown by arrows in Figure 4.
By definition.
Mybius's group of transformations preserve the values of angles and transform spheres into spheres.
In addition.
Mybius's transformations are locally similar transformations, as shown in Figure 5.
Figure 4.
Correspondence of the plane U1 ,U 2 , .
: to conformal plane n1 , n2 , .
: for U1 A const Figure 5.
Growth transformations in cap of fungus and their modeling as Mybius's transformations.
GROWTH CHANGES OF VALUES OF THE HUMAN BODY AS NON- EUCLIDEAN
TRANSFORMATIONS
Nonlinear transformations of human skull as a function of aging are described as Mybius's transformations in Figure 6.
Growth changes of three-component kinematic blocks .
he phalanxes of fingers, the threemembered extremities and the three- membered bod.
are characterized by the constant value of the crossratio of four points.
In particular, for the fingers the cross- ratio has is given by:
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(C A A) E ( D A B )
A const
(C A B ) E ( D A A)
The expressions in parentheses are the lengths of segments between the end points of finger AB.
BC, and CD are the lengths of basic phalanx, middle one and the end phalanxes.
The values of the cross-ratio of all the blocks, at least during individual development, are grouping around the Meanwhile, the growth of the human body is essentially nonlinear, as shown in Figure 7.
Figure 6.
Mybius's transformations in the modelling of ontogenetic transformations of human skulls.
Profiles of the skull of, .
: an adult, and .
: a 5-year old child.
Thus, all three-membered blocks of the human kinematics are Mybius equivalent and Mybius invariable during the human lifetime.
Figure 7.
Changes of the growing human body with, .
: aging, and .
: three-stretch parts cross- ratios are equal to1.
On the Analogy of Non-Euclidean Geometry of Human Body With Electrical Networks (Ashok Vaseasht.
A ISSN: 2088-8708
PROJECTIVE TRANSFORMATIONS OF ELECTRIC NETWORK - THE CROSS- RATIO
VALUES
Let an electric circuit .
n active two-pole circuit A), as shown in Figure 8 is considered.
At change of load RL> 0 from a regime of short circuit (SC) (RL = .
to the open circuit (OC) (RL= C.
, the load straight line or I-V characteristic I L (U L ) is obtained.
Further, it is possible to calibrate the I-V characteristic by the load resistance values variation.
The equation VL ( RL ) has the characteristic linear-fractional view.
VL A V0 Ri A RL Figure 8.
Electric circuit with variable load and its I-V characteristic It gives a basis for consideration of the transformation of the straight line RL into line VL as projective one in Figure 9.
Figure 9.
Projective transformation R L CU L and the cross ratio mL It is convenient to use the points of characteristic regimes, as the pairs of respective points, namely the short circuit, open circuit, and maximum load power.
If the fourth point is the point of running regime.
RL1 .
VL1 .
I 1L , then the cross ratio mL has the form, m1L A .
RL1 Ri Cu ) A RL1 A 0 Ri A 0 RL1 R1L A Cu Ri A Cu Ri IJECE Vol.
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m1L A ( 0 VL1 0,5V0 V0 ) A A ISSN: 2088-8708
VL1
V0 A VL1
Thus, the coordinate of running regime point is a set of this values mL , which is defined in the invariant manner through the various regime parameters.
RL .
VL .
The regime change RL C RL can be expressed similarly as, mL21 A .
RL2 RL1 Cu ) A RL2
VL2
VL1
RL1 V0 A VL2 V0 A VL1
Now, it should be made the identical changes of the regime for different initial regimes on the line VL in Figure 10.
Figure 10.
Identical changes of regime for different initial regimes For this purpose, from the equation .
we obtained an expression for VL ( VL ) , eliminating for two values RL2 .
R1L
VL1
VL2
L21 A .
E L A 1
mL21 E The transformation obtained using the parameter mL translates the point of initial regime VL into the point VL .
Therefore, by keeping the parameter of this transformation invariable and by setting different values of initial regime VL11 .
VL12 , etc.
, we obtain the points of the subsequent regimes VL21 ,VL22 , etc.
, which form a segment of invariable length .
n sense of projective geometr.
, that is considered as a segment movement in geometry.
Here, the character of a change of Euclidean .
length of the segment is visible.
Approaching to the base points.
Euclidean length is decreasing to zero and then is increasing again at the moment of transition to the external area.
Thus, regime changes are projectively similar for different initial The change of a segment with an invariable value of the cross-ratio is similar to that shown in Figure 7.
In the theory of the projective transformations, the fixed points play an important role, which can be considered as the base points.
For their finding, the equation .
is solved for condition VL1 A VL2 .
It turns out the two real roots.
VL A 0.
VL A V0 which define a hyperbolic transformation and hyperbolic (Lobachevsk.
geometry, respectively.
If roots of the equation coincide, one fixed point defines a parabolic transformation and, respectively, parabolic (Euclidea.
If roots are imaginary, geometry is elliptic (Riemannia.
Input- output projective conformity of network - Let us consider a two-port circuit TP (Figure .
It can be represented in an equation, such as.
On the Analogy of Non-Euclidean Geometry of Human Body With Electrical Networks (Ashok Vaseasht.
A ISSN: 2088-8708
EV0
E E ch A sh A EE
EE A EE
E I 0 E A E E sh A ch A E
E E V1
EE E EE
E E I1 E A E
where A is an attenuation coefficient.
A is characteristic or wave resistance.
This transformation can be seen as a rotation of the radius-vector 0YL of constant length at the angle A to the position 0YIN in the pseudo-Euclidean space I .
V (Figure 11.
Then, we have also the following invariant V 1 2 A I 12 A 2 A V 02 A I 02 A 2 as length of the vector 0YL .
This approach corresponds to Lorenz transformations in mechanics of the relative motion.
The conductivities at the input and output of the two-port network are connected already by linear- fractional expression.
YIN1A A YL1A A thA 1 A YL1A E thA This expression corresponds to the rule of addition of relativistic velocities.
Let us consider the points YL1 .
YIN 1 on superposed axis in Fig.
This figure represents a point movement from the position YL1 to the position YIN 1 .
r the segment movement YL1 YIN 1 ) for different initial values YL1 as shown by arrows.
Then, the points C 1 are fixed.
The cross-ratio of the points YL1 .
YIN 1 , relatively fixed points, determines the AulengthAy of segment YL1 YIN 1 or the maximum of efficiency K P M m (Y L1 Y IN 1 ) A of a two-port circuit:
1 A th A A KPM 1 A th A Thus, one more founded invariant is equal to a concrete number.
That invariant is similar to the cross-ratio of the human body equal to 1.
Figure 11.
: Two-port network .
: input- output characteristic.
: movement of the segment for different initial values .
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June 2014 : 378 Ae 388
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ISSN: 2088-8708
Cascade connection of two-ports - Let us consider the cascaded two-ports TP1 and TP2 in Figure The relationship of regime parameters at different parts of the network or AumovementAy on these parts also corresponds to projective transformations.
The load change from the value YL 2 to the value YL 2 defines the correspond changes YL1 .
YIN1 .
The length of segments of all the load lines is different for the usually used Euclidean geometry.
If the mapping is viewed as the projective transformation, the invariant, which is the cross-ratio of four points, is performed and defines the same length of segments.
Thus, networks of this kind are projectively Ae similar.
Therefore, there is some kind of the electro - biological analogy: disproportionate change of segments of load line for different parts of a network corresponds to growth changes of parts of the human body.
Projective plane - If the network containing two changeable loads, projective geometry for the plane can be shown.
In that case, the load straight lines form the coordinate triangles.
Therefore, networks of this kind are also projectively - similar.
Figure 12.
: Cascade connection of two two-ports.
: Corresponding I-V characteristics.
CONFORMAL TRANSFORMATIONS OF ELECTRIC NETWORK
Let us consider a power supply system with two voltage regulators VR1 .
VR2 and loads R1 .
R2 shown in Figure 13.
The regulators define voltage transmission coefficient or transformation ratio n1 , n2 .
The voltage regulators are connected to a limited capacity supply, voltage source U 0 .
An interference of the regulators on regimes or load voltages U1 .
U 2 is observed because of existence of an internal resistance Ri Let us consider the case Ri A R1 A R2 .
Figure 13.
Power supply system with two voltage regulators On the Analogy of Non-Euclidean Geometry of Human Body With Electrical Networks (Ashok Vaseasht.
A ISSN: 2088-8708 The network behavior or AukinematicsAy via variable parameters n1 , n2 is described by a sphere in the coordinates U 1 ,U 2 ,U i in Figures 3, 4.
For regulation, it is better to use such groups of transformations or movements of points in the planes U1 ,U 2 and n1 , n2 , when it is impossible to deduce a working point over the circles, which correspond to the equator of sphere by a finite switching number.
In this sense, we derive a hyperbolic geometry.
On the plane U1,U 2 it is the Beltrami-Klein's model and on the plane n1 , n2 it is the Poincare's model.
The corresponding circle carries the name of the absolute and defines an infinitely remote border.
Let us put the value n2 A 0 .
Then, the regime change goes only on axes U1 and n1 .
The conformity of the characteristic points and running point is shown in Figure 14.
Figure 14.
Conformity of the variables U1 , n1 of the hyperbolic transformations From the methodical point of view, it is useful to consider the hyperbolic transformation by the analogy, corresponding to the relativistic rule of speed composition in relative movement mechanics.
If, for example U 1 A 0.
5 , then U1 A 0.
5 and it does not depend from the value U1 .
In the case Ri C R1 C R2 ,the sphere will be transformed to an ellipsoid.
Therefore, these network will be conformally or Mybiussimilar for the plane n1 , n2 .
Therefore, there is some kind of the electro - biological analogy: the change of network parameters corresponds to growth changes of biological objects.
CONCLUSION
The analysis of human growth and analysis of operating regimes of electric networks shows an invariant of projective and conformal transformations.
Different types of the cross-ratio take place for an electric network.
The change of an operating regime of the given network or change of network parameters results to the projective or conformal similarity of networks.
The established electro-biological analogy develops a methodological basis of application of non-Euclidean geometries for these areas.
REFERENCES