Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol
.
4
,
No
. 3,
J
une
2
0
1
4
,
pp
. 37
8~
38
8
I
S
SN
: 208
8-8
7
0
8
3
78
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
On the Analogy of Non-Euclid
ean Geometry of Human Body
With Electrical Networks
Alexander
Pe
nin*,
Anatoli
Si
doren
ko*,
Ashok
Vase
ash
t
a**
* D. Ghitu Institute of
Electronic Engin
eer
ing
an
d Nanotechnolo
g
ies, Ch
isinau
,
Moldova
** IASC/ICWI,
NUARI, 13873
Park Cent
er Rd
.
Ste 500, Herndo
n VA, USA
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ja
n 13, 2014
Rev
i
sed
Ap
r
20
, 20
14
Accepte
d
May 8, 2014
A review of ap
plication of non
-Eucli
dean g
e
o
m
etries for interpreting th
e
process of th
e gr
owth in th
e human bod
y is
pr
es
e
n
ted and
fe
ature
s
em
plo
y
in
g
non-Euclidean
g
e
ometries in the
electr
i
c
circuit theor
y
ar
e modeled. Growth
of the human bo
d
y
and ch
anges
of para
meters of an oper
a
ting r
e
gime of an
electronic netwo
r
k correspond to projec
tiv
e and
conformal transformations
which possess an invarian
t bein
g the cr
oss-ratio
of four points.
The common
m
a
them
atic
al
a
pparatus r
e
pres
ents int
e
rdisciplinar
y
approach
in view of
analog
y
of processes of a diff
erent ph
y
s
ical n
a
tur
e
.
The r
e
sults ob
tain
ed her
e
demonstrate d
e
v
e
lopment of
a m
e
thodolog
y
of
application
of no
n-Euclidean
geometries and its
biological corr
ela
tion
to th
e gro
w
th of human bo
d
y
.
Keyword:
Electrical network
H
u
m
a
n
bod
y
M
öbi
us e
qui
va
l
e
nt
No
n-
Eucl
dean
geom
et
ry
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Ashok Vaseas
hta,
IASC/IC
W
I
,
N
UAR
I,
1
387
3 Par
k
Cen
t
er
R
d
.
Su
ite
5
00
Her
n
do
n,
V
A
US
A
Em
a
il: prof.va
s
eashta@ieee.org
1.
INTRODUCTION
The g
r
o
w
t
h
o
f
t
h
e hum
an bo
dy
i
s
essen
t
i
a
l
l
y
a nonl
i
n
ear p
r
ocess fo
r or
di
na
ry
or E
u
cl
i
d
ea
n
geom
etry. Ana
l
ysis of the
structure
and growth reveals
t
h
at
separate bod
y pa
rts or all
the three-c
o
mpone
n
t
k
i
n
e
m
a
tics
b
l
o
c
ks (th
e
ph
alan
x
e
s o
f
fi
n
g
e
rs, th
e th
r
ee-m
e
m
b
ered
ex
tremities an
d
th
e t
h
ree-m
e
m
b
ered
bo
d
y
)
change
accordi
n
g to Möbius
trans
f
orm
a
tions which a
r
e c
h
a
r
acteristic for c
o
nform
a
l
and projective ge
ometry.
Suc
h
t
r
ans
f
or
m
a
t
i
ons pos
se
ss an i
nva
ri
ant
or i
n
vari
ab
le value.
T
h
erefore,
all
the above m
e
ntioned three-
com
pone
nt
bl
o
c
ks are c
h
aract
eri
zed
by
a con
s
t
a
nt
val
u
e t
h
r
o
u
g
h
o
u
t
t
h
e
hu
m
a
n l
i
f
e [1]
,
[
2
]
.
From
t
h
i
s
po
i
n
t
o
f
view, E
u
clidea
n ge
om
etry appears as
a
pos
sible analytical instrum
e
nt. The
adve
nt of the special theory of
relativ
ity led
to
a n
e
w term
, “g
eo
m
e
trizati
o
n of
ph
ys
i
c
s”, w
h
i
c
h
i
s
a
m
e
t
h
o
dol
ogi
cal
doct
r
i
n
e a
n
d
m
a
y
be
defi
ned a
s
t
h
e
appl
i
cat
i
o
n o
f
geom
et
ri
cal
m
e
t
h
o
d
s i
n
p
h
y
s
ics, wh
en
ev
er
p
o
s
sib
l
e. Th
is
is form
al
ly th
e th
eory
of
i
n
vari
a
n
t
s
o
f
s
o
m
e
gro
u
p
of
t
r
a
n
sf
orm
a
t
i
ons
(
P
oi
ncaré
-
Lore
nt
z
gr
o
u
p
)
, o
r
space
-t
i
m
e ge
om
et
ry
. Th
e basi
c
pri
n
ciples of projective ge
om
etry which a
r
e
reflected
in
meth
od
s of lin
ear p
e
rsp
ective
.
The
ge
om
aterization
of
phy
si
cs,
bi
ol
o
g
y
,
an
d ne
ur
osci
ence c
o
vere
d a m
u
l
t
i
t
ude
of fi
el
d
s
and as a res
u
l
t
of i
t
s
bi
ol
ogi
cal
application, there is an im
age of a
su
bject
i
n
whi
c
h
bot
h di
st
ances an
d co
rne
r
s cha
n
ge. I
t
i
s
im
port
a
nt
t
o
n
o
t
e
that these c
h
a
nge
s a
ppea
r
neither ar
b
itr
ar
ily o
r
r
a
ndo
m
l
y, bu
t ar
e invar
i
an
t
b
e
ing
the cro
ss- ratio
o
f
fo
ur
poi
nt
s
on
a st
ra
i
ght
l
i
n
e
(a
d
o
u
b
l
e
p
r
op
ort
i
o
n
)
.
Geom
etry of
the Euclidea
n space com
p
le
m
e
nted by
one infi
nitely re
m
o
te point, i
s
called the
co
nfo
r
m
a
l g
e
ometry [3
]. Projectiv
e in
terp
ret
a
tio
n
in
t
h
e
fo
rm
of a st
ere
o
gra
p
hi
c p
r
o
j
ect
i
on
of
p
o
i
n
t
s
o
f
t
h
e
sphere to t
h
e
plane gives an e
x
am
pl
e of the
conform
a
l plane (as an exam
pl
e of s
u
ch
st
ere
o
g
r
a
phi
c
pr
o
j
e
c
t
i
o
n
i
s
cart
o
g
r
a
phy
)
,
w
h
en t
h
e val
u
e o
f
t
h
e cr
oss
-
rat
i
o
of f
o
ur
poi
nt
s rem
a
i
n
s const
a
nt
[
4
]
.
Suc
h
pl
a
n
e p
r
o
j
ect
i
o
n
is app
lied
in physics f
o
r
so
lv
i
n
g
o
f
electr
o
stat
ic p
r
ob
lem
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
37
8 – 3
8
8
3
79
M
öbi
us'
s
cert
a
in su
b
g
r
o
u
p
o
f
t
r
ans
f
o
r
m
a
ti
on
s cor
r
esp
o
n
d
s
t
o
Lo
bac
h
ev
ski
’
s ge
om
et
ry
or hy
per
b
ol
i
c
geom
et
ry
, t
h
e so-cal
l
e
d P
o
i
n
c
a
re'
s
conf
orm
a
l
i
n
t
e
rpret
a
t
i
o
n
[5]
.
It
i
ndi
cat
e
s
t
h
at
t
h
e space
of vi
s
u
al
perc
ept
i
on
is cha
r
acterize
d
by L
oba
che
v
ski
ge
om
etry. A num
b
er
of
publications s
h
ow t
h
at in el
ectric networks, the
chan
ges
of
op
erat
i
ng
regi
m
e
param
e
t
e
rs can be i
n
t
e
rp
r
e
ted
as pr
oj
ectiv
e an
d
co
nf
orm
a
l tran
sfo
r
m
a
tio
n
s
.
In
ad
d
ition
,
th
e relatio
n
s
h
i
p
o
f
reg
i
m
e
p
a
ram
e
ters
at d
i
ffe
rent p
a
rts of a
n
e
twork also is
b
e
ing
d
e
scrib
e
d
b
y
p
r
oj
ectiv
e tran
sfo
r
m
a
tio
n
s
[6
], [7
], [8
]. An
in
te
rdisciplinary approac
h
ap
p
lies commo
n
m
a
th
ematica
l
appa
rat
u
s
i
n
v
a
ri
o
u
s area
s o
f
sci
e
nce a
n
d
t
h
e si
m
i
l
a
ri
t
y
of
p
r
oces
ses
o
f
di
ffe
rent
phy
si
cal
nat
u
re
.
As t
h
e
exam
ple of s
u
ch inte
rdisci
pl
i
n
ary
a
p
pr
oach
,
we
p
r
esent
a
r
e
vi
ew
o
f
a
ppl
i
cat
i
on
of
n
o
n
-
E
ucl
i
d
ea
n
geo
m
et
ri
es
f
o
r
in
terp
r
e
tati
o
n
of
th
e
h
u
m
an
bod
y gr
ow
t
h
pro
cess, and
the
features of use of n
on-Euclidean geom
etries in
the electric circuit theory a
r
e
shown.
2.
ON T
H
E FEATURES
OF
G
E
OMETRI
C
TRANSF
ORMATI
ONS
Projec
tive tr
ansformations General
c
a
se
:
Gen
e
rally, th
e
p
r
oj
ectiv
e transform
a
tio
n
o
f
p
o
i
n
t
s of on
e
straig
h
t
lin
e
L
U
into
th
e po
in
ts of th
e
o
t
h
e
r lin
e
L
R
i
s
set
o
f
poi
nt
s
wi
t
h
t
h
e
pr
oj
ec
t
i
on ce
nt
er
S
or the three
pai
r
s of respec
t
i
v
e
poi
nt
s
(s
h
o
w
n
i
n
Fi
g
u
re 1)
.
The pr
o
j
ect
i
v
e
t
r
ans
f
orm
a
ti
ons prese
r
ve
t
h
e
cr
oss rat
i
o
of fo
ur
poi
nt
s,
4
3
1
3
4
2
1
2
4
3
2
1
)
(
L
L
L
L
L
L
L
L
L
L
L
L
R
R
R
R
R
R
R
R
R
R
R
R
m
)
(
4
3
2
1
L
L
L
L
U
U
U
U
m
Fig
u
re
1
.
Proj
ectiv
e tran
sfo
r
m
a
tio
n
o
f
straight lin
es po
in
ts
Af
fine tra
n
sf
o
rmati
on
:
T
h
ere is the
projec
tion ce
nter
S,
b
u
t
th
e
straight lin
es are
L
L
R
U
,
p
a
rallel.
There
f
ore,
t
h
e
i
nva
ri
ant
of
an
affi
ne
t
r
a
n
sf
or
m
a
t
i
on i
s
t
h
e
si
m
p
l
e
rat
i
o
or
p
r
o
p
o
rt
i
o
n
of
t
h
ree
poi
nt
s.
Euclidean transformation:
If t
h
e
projection
S
cen
ter an
d th
e strai
g
h
t
lin
es
L
L
R
U
,
are
parallel, the
proj
ection is carried
out by
parallel
lines. This
proj
ection corr
es
ponds to t
h
e Euc
lidean
transform
a
t
i
on, which is
parallel tr
anslation
of a
segm
ent. T
h
e E
u
c
lidean transformation preserves t
h
e
difference of poi
nts. The tr
ansform
a
t
i
ons of an i
n
itial
rectangular
coordinate grid, consi
d
ered here are
prese
n
ted
in Figu
re 2.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
On
t
h
e
An
al
o
g
y
of
N
o
n
-
E
u
cl
i
d
ea
n
Ge
omet
ry
of
Hu
m
a
n
B
o
d
y
Wi
t
h
El
ect
ri
cal
N
e
t
w
orks (
A
sho
k
V
a
se
asht
a)
38
0
Stere
ogr
aphic projecti
o
n:
T
h
e pr
oj
ect
i
o
n
of p
o
i
n
t
s
of
t
h
e
sp
here
)
,
(
2
1
U
U
U
i
from
t
h
e t
o
p
pol
e
o
n
t
h
e t
a
nge
nt
pl
a
n
e
2
1
,
n
n
placed at the bottom
pole
is prese
n
ted in Fi
gure 3. For
the sake
of si
m
p
licity,
the
coo
r
di
nat
e
a
x
e
s
2
1
,
U
U
and
2
1
,
n
n
are c
o
inc
i
dent.
Figure
2. a): C
h
aracteristic transf
or
m
a
t
i
o
n
of
th
e Car
t
esian
g
r
i
d
b
y
v
a
r
i
ou
s gr
oup
tr
an
sformatio
n
s
,
b
)
:
Eu
clid
ean
,
c):
affin
e
, an
d d):
p
r
oj
ectiv
e.
Fi
gu
re
3.
St
ere
o
g
r
a
phi
c
pr
o
j
e
c
t
i
on
of
t
h
e s
p
here
o
n
t
h
e c
o
n
f
o
r
m
a
l
pl
ane
2
1
,
n
n
The c
o
nf
orm
a
l
pl
ane
di
ffe
r
s
f
r
om
Eucl
i
d
ean
by
exi
s
t
e
nce
of
o
n
e
i
n
fi
ni
t
e
l
y
rem
o
t
e
poi
nt
co
rresp
ond
ing
to
th
e t
o
p po
le
o
f
th
e sph
e
re.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
37
8 – 3
8
8
3
81
Co
nf
orm
a
l
tr
ans
f
orm
a
ti
ons
:
T
h
e a
r
ea
of
c
h
anges
of the
values
2
1
,
U
U
corresponds
to the s
p
he
re
equat
o
r
i
n
Fi
g
u
r
e 4a.
Let th
e
v
a
lu
e
1
U
t
o
b
e
,
const
U
1
(th
a
t i
s
th
e lin
e
1
L
).
The circular section
2
L
i
s
on t
h
e
s
p
here
wh
ile th
e ci
rcle
3
L
i
s
on t
h
e
pl
an
e
2
1
,
n
n
. Th
e
similar fam
i
ly o
f
circles is
desc
ri
be
d
by
t
h
e
rot
a
t
i
o
n g
r
ou
p
of
sp
here, as it is shown b
y
arro
ws in Figu
re
4
.
By d
e
fin
itio
n, M
ö
b
i
u
s
's group
o
f
transform
a
t
i
o
n
s
p
r
eserv
e
th
e
val
u
es
of a
ngl
e
s
and t
r
a
n
s
f
o
r
m
spheres i
n
t
o
sphe
res.
In
a
d
di
t
i
on, M
öbi
us'
s
t
r
ansf
o
r
m
a
ti
ons are l
o
cal
l
y
sim
i
l
a
r
t
r
ans
f
o
r
m
a
ti
on
s, as s
h
ow
n i
n
Fi
gu
re
5.
Fi
gu
re 4.
C
o
rre
spo
n
d
e
n
ce of
t
h
e pl
ane
2
1
,
U
U
, a): t
o
conform
a
l plane
2
1
,
n
n
, b): fo
r
const
U
1
Fi
gu
re
5.
G
r
o
w
t
h
t
r
a
n
sf
orm
a
t
i
ons
i
n
cap
o
f
f
u
n
g
u
s a
n
d t
h
ei
r m
odel
i
n
g
as
M
öbi
us'
s
t
r
ans
f
o
r
m
a
ti
ons.
3.
GRO
W
TH CHANGES O
F
VAL
UES
OF THE
HU
MAN
BODY
AS N
O
N- EU
C
L
ID
EAN
TRANSFORMATIONS
No
nl
i
n
ea
r t
r
a
n
sfo
r
m
a
t
i
ons o
f
h
u
m
a
n skul
l
as a f
u
nct
i
o
n
of a
g
i
n
g a
r
e desc
ri
be
d a
s
M
ö
bi
us'
s
tran
sform
a
t
i
o
n
s in
Fi
gu
re
6
.
Gr
owt
h
cha
n
g
e
s of t
h
ree
-
c
o
m
pone
nt
ki
n
e
m
a
t
i
c
bl
ocks
(t
he p
h
al
an
x
e
s of
fi
n
g
ers
,
t
h
e t
h
ree
-
me
m
b
ered
ex
tre
m
it
ies an
d the th
ree-
m
e
m
b
ered body) are
cha
r
acterized
by
th
e con
s
tant v
a
lu
e
of th
e
cro
s
s-
ratio
o
f
fo
ur p
o
in
ts.
In
p
a
rticular,
for
th
e
fi
n
g
e
rs t
h
e
cr
oss
-
r
a
t
i
o
has
i
s
gi
ve
n
by
:
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
On
t
h
e
An
al
o
g
y
of
N
o
n
-
E
u
cl
i
d
ea
n
Ge
omet
ry
of
Hu
m
a
n
B
o
d
y
Wi
t
h
El
ect
ri
cal
N
e
t
w
orks (
A
sho
k
V
a
se
asht
a)
38
2
co
nst
A
D
B
C
B
D
A
C
W
)
(
)
(
)
(
)
(
The e
x
p
r
essi
o
n
s i
n
pa
rent
he
ses are t
h
e l
e
ngt
hs
of se
gm
ent
s
bet
w
een
t
h
e en
d
poi
nt
s
of
fi
n
g
e
r
phal
a
nxe
s. AB
, B
C
,
and C
D
are t
h
e l
e
n
g
t
h
s
of ba
si
c p
h
al
anx
,
m
i
ddl
e one
and t
h
e e
n
d p
h
al
an
xes. T
h
e
val
u
e
s
of t
h
e cr
oss-
r
a
t
i
o
of al
l
t
h
e bl
oc
ks, at
l
east
du
ri
n
g
i
n
di
vi
d
u
al
de
vel
opm
ent
,
are
g
r
o
u
p
i
n
g ar
ou
n
d
t
h
e
b
e
n
c
h
m
ar
k
1
.
31
. Meanw
h
ile,
th
e gro
w
t
h
o
f
t
h
e
h
u
m
an
body is essen
tially n
o
n
lin
ear
,
as sh
own
in Figur
e 7
.
Fi
gu
re
6.
M
ö
bi
us'
s
t
r
ans
f
o
r
m
a
t
i
ons i
n
t
h
e
m
odel
l
i
ng
o
f
ont
o
g
enet
i
c
t
r
a
n
s
f
o
r
m
a
ti
ons
of
h
u
m
an sk
ul
l
s
. P
r
ofi
l
e
s
o
f
th
e sku
ll of,
a): an adu
lt,
a
n
d
b):
a
5
-
y
ear
o
l
d chi
l
d
.
Th
us, al
l
t
h
re
e-m
e
m
b
ered b
l
ocks
o
f
t
h
e
h
u
m
a
n ki
nem
a
ti
cs are M
öbi
u
s
eq
ui
val
e
nt
a
n
d
M
ö
bi
us
i
nva
ri
abl
e
du
ri
ng
t
h
e h
u
m
a
n
l
i
f
et
im
e.
Fi
gu
re
7.
C
h
a
n
ges
of
t
h
e
g
r
o
w
i
n
g
hum
an b
ody
wi
t
h
,
a):
a
g
i
n
g, a
n
d
b):
t
h
ree-st
ret
c
h
part
s cr
oss-
rat
i
o
s
are
eq
u
a
l t
o
1.31
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
37
8 – 3
8
8
3
83
4.
PROJ
ECTIVE TRANSF
ORMATIONS OF
ELECT
R
IC NETWORK
-
THE CROSS- RATIO
VAL
UES
Let an electric
circuit (a
n active two-pole circuit A)
,
as s
h
ow
n i
n
Fi
g
u
r
e
8 i
s
c
onsi
d
ere
d
. At
c
h
a
n
g
e
of l
o
a
d
R
L
> 0 from
a regim
e
of s
h
o
r
t circuit (SC) (R
L
= 0) t
o
t
h
e ope
n ci
r
c
ui
t
(OC
)
(R
L
=
), th
e lo
ad
st
raigh
t
lin
e o
r
I-V ch
aracteristic
)
(
L
L
U
I
is ob
tain
ed. Fu
rt
h
e
r, it is po
ssib
l
e to
calib
rate the I-V ch
aracteristic b
y
th
e
load resistance
value
s
variation.
Th
e equ
a
tio
n
)
(
L
L
R
V
h
a
s th
e ch
aracteristic lin
ear-fractio
n
a
l v
i
ew,
L
i
L
L
R
R
R
V
V
0
Figu
re 8.
Electric
circuit with vari
ab
le lo
ad
an
d its I-V ch
aracteristic
It
gi
ves
a bas
i
s fo
r co
nsi
d
e
r
at
i
on
of
t
h
e t
r
ans
f
orm
a
t
i
on of t
h
e st
rai
g
ht
l
i
n
e R
L
in
to
lin
e V
L
as
pr
o
j
ect
i
v
e one
i
n
Fi
g
u
r
e 9.
Fi
gu
re
9.
Pr
o
j
e
c
t
i
v
e t
r
ans
f
orm
a
t
i
on
L
L
U
R
and t
h
e c
r
oss
ratio
L
m
It is co
nv
en
ient to
u
s
e th
e
po
in
ts of ch
aracteristic
reg
i
m
e
s, as th
e p
a
i
r
s of
resp
ectiv
e po
ints, n
a
m
e
ly
th
e sh
ort circuit, o
p
en circu
it, and m
a
x
i
m
u
m
lo
ad
po
wer.
If th
e fou
r
th
p
o
i
n
t
is th
e
p
o
in
t of
runn
ing
reg
i
m
e
,
1
1
1
,
,
L
L
L
I
V
R
, th
en
th
e cro
s
s ratio
L
m
has
the
f
o
rm
,
i
L
i
i
L
L
i
L
L
R
R
R
R
R
R
R
R
m
1
1
1
1
1
0
:
0
)
0
(
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
On
t
h
e
An
al
o
g
y
of
N
o
n
-
E
u
cl
i
d
ea
n
Ge
omet
ry
of
Hu
m
a
n
B
o
d
y
Wi
t
h
El
ect
ri
cal
N
e
t
w
orks (
A
sho
k
V
a
se
asht
a)
38
4
1
0
1
0
0
1
1
)
5
,
0
0
(
L
L
L
L
V
V
V
V
V
V
m
(1
)
Thus,
t
h
e coo
r
d
i
na
te o
f
runnin
g
reg
i
m
e
p
o
i
n
t
is
a
set
o
f
t
h
is va
lu
es
L
m
, which
is d
e
fi
n
e
d in
th
e
i
nva
ri
ant
m
a
n
n
er t
h
ro
u
gh t
h
e vari
o
u
s re
gi
m
e
param
e
t
e
rs,
L
L
V
R
,
. The regim
e
change
2
1
L
L
R
R
can be
expresse
d sim
i
larly as,
1
0
1
2
0
2
1
2
1
2
21
:
)
0
(
L
L
L
L
L
L
L
L
L
V
V
V
V
V
V
R
R
R
R
m
(3
)
Now, it shou
ld b
e
m
a
d
e
th
e id
en
tical ch
anges of th
e reg
i
me fo
r d
i
fferen
t
in
itial reg
i
m
e
s on
t
h
e lin
e
L
V
in
Figur
e
1
0
.
Fig
u
re
10
.
Id
entical ch
ang
e
s
of reg
i
m
e
fo
r
d
i
fferen
t in
itial reg
i
m
e
s
For
t
h
i
s
p
u
r
p
os
e, f
r
om
t
h
e eq
u
a
t
i
on
(2
)
we
ob
t
a
i
n
ed a
n
e
x
pre
ssi
on
f
o
r
)
(
1
2
L
L
V
V
, elimin
atin
g
i
R
fo
r tw
o
value
s
1
2
,
L
L
R
R
1
)
1
(
0
1
21
0
1
21
0
2
V
V
m
V
V
m
V
V
L
L
L
L
L
(4
)
The t
r
a
n
s
f
o
r
m
a
t
i
on
obt
ai
n
e
d
usi
n
g t
h
e p
a
ra
m
e
t
e
r
21
L
m
tran
slat
es th
e
p
o
i
n
t
o
f
in
itial reg
i
m
e
1
L
V
in
t
o
th
e po
in
t
2
L
V
. Ther
efo
r
e, by
keepi
ng t
h
e param
e
t
e
r of t
h
i
s
t
r
ans
f
o
r
m
a
ti
on i
n
va
ri
abl
e
an
d by
s
e
t
t
i
ng di
f
f
ere
n
t
v
a
lu
es of i
n
itial reg
i
m
e
1
2
1
1
,
L
L
V
V
, etc., we ob
tain
t
h
e po
in
ts
o
f
th
e
su
bsequ
e
n
t
regi
m
e
s
2
2
2
1
,
L
L
V
V
, etc.,
which
fo
rm
a segm
ent
o
f
i
n
va
ri
abl
e
l
e
ngt
h
(i
n s
e
nse
of
pr
oje
c
t
i
v
e ge
om
et
r
y
), t
h
at
i
s
c
o
n
s
i
d
ere
d
as
a s
e
gm
ent
m
ove
m
e
nt in geom
etry. Here, the c
h
aracte
r
of a c
h
ange
of
Eu
clid
ean
(u
sual) leng
th
of the seg
m
en
t is v
i
sib
l
e.
Ap
pr
oac
h
i
n
g t
o
t
h
e base
poi
nt
s, Eucl
i
d
e
a
n
l
e
ngt
h i
s
decr
easi
ng t
o
zer
o
and t
h
e
n
i
s
i
n
creasi
n
g agai
n
at
t
h
e
m
o
men
t
o
f
tran
sitio
n
t
o
th
e
ex
tern
al area.
Thu
s
, reg
i
m
e
c
h
ang
e
s are proj
ectiv
ely si
m
i
l
a
r fo
r
d
i
fferen
t
in
itial
reg
i
m
e
s. Th
e ch
ang
e
o
f
a seg
m
en
t with
an
inv
a
riab
le
valu
e o
f
t
h
e cro
ss-ratio
is similar to
th
at s
h
own
in
Figu
re 7
b
).
In
t
h
e th
eory of th
e
proj
ectiv
e tran
sform
a
tio
n
s
, t
h
e
fi
xe
d
p
o
i
n
t
s
pl
ay
an i
m
port
a
nt
r
o
l
e
,
whi
c
h ca
n
be
co
nsid
ered
as t
h
e
b
a
se
p
o
i
n
t
s. For th
ei
r fi
n
d
i
n
g
, th
e equ
a
tion
(4) is so
lv
ed for con
d
ition
2
1
L
L
V
V
. It tur
n
s
out
the two real
roots,
0
,
0
V
V
V
L
L
whi
c
h
de
fi
ne a
hy
pe
rb
o
l
i
c
t
r
ansf
orm
a
ti
on a
n
d hy
per
bol
i
c
(
L
o
b
ac
he
vski
)
geom
et
ry
, resp
ect
i
v
el
y
.
If r
o
o
t
s of t
h
e e
q
uat
i
on c
o
i
n
ci
de,
o
n
e fi
xed
p
o
i
n
t
defi
nes a pa
ra
bol
i
c
t
r
a
n
sf
or
m
a
t
i
on
an
d,
resp
ectively, p
a
rabo
lic (Eu
c
lid
ea
n
)
g
e
ometry. If
roo
t
s
are im
ag
in
ary,
g
e
o
m
etry is ellip
tic (Riem
a
n
n
i
an
).
Inpu
t- output projective
c
o
n
f
ormi
ty of
ne
twork
-
Let us
consider a two-port ci
rc
uit TP (Fig
ure
1
1
).
It can
be
rep
r
ese
n
t
e
d
i
n
an
eq
uat
i
o
n,
suc
h
as
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
37
8 – 3
8
8
3
85
1
1
0
0
I
V
ch
sh
sh
ch
I
V
whe
r
e
is an
atten
u
a
tion
co
effi
cien
t,
is ch
aracteristic o
r
wav
e
resistan
ce. Th
is tran
sformatio
n
can
be see
n
as
a rotation
of the radi
us-vect
or
L
Y
0
o
f
c
onst
a
nt
l
e
ngt
h at
t
h
e
an
g
l
e
to
th
e
po
sitio
n
IN
Y
0
in
the pseudo-E
uclidean s
p
ace
V
I
,
(Fi
g
ure
1
1
b
)
.
The
n
, we ha
ve al
so
t
h
e
f
o
l
l
o
w
i
ng
i
nva
ri
ant
2
2
0
2
0
2
2
1
2
1
I
V
I
V
as length
of the vector
L
Y
0
.
This
approach corre
sponds t
o
L
o
renz
tran
sform
a
tio
n
s
i
n
m
ech
an
ics
of
th
e relativ
e m
o
tio
n
.
Th
e con
ductiv
ities at th
e in
pu
t and
ou
tpu
t
of th
e two
-
po
rt
n
e
two
r
k
are con
n
ected
al
read
y
by linear-
fract
ional e
x
pressi
on,
th
Y
th
Y
Y
L
L
IN
1
1
1
1
Th
is ex
pressi
on
co
rresp
ond
s
to
th
e
ru
le of
ad
d
ition
o
f
rel
a
tiv
istic v
e
lo
cities. Let
u
s
co
n
s
i
d
er the
poi
nt
s
1
1
,
IN
L
Y
Y
o
n
superpo
s
ed
ax
is in Fig
.
11
(c). Th
i
s
figu
re
rep
r
esen
ts a
p
o
i
n
t
mo
v
e
m
e
n
t
fro
m
th
e po
sition
1
L
Y
to
th
e po
sitio
n
1
IN
Y
(
o
r
t
h
e
segm
ent
m
ovem
e
nt
1
1
IN
L
Y
Y
) fo
r
d
i
fferen
t i
n
itial v
a
lu
es
1
L
Y
as show
n
by arrows
. T
h
en, t
h
e
points
1
are
fix
e
d
.
The cro
ss-ratio
of th
e po
in
ts
1
1
,
IN
L
Y
Y
, relativ
ely fix
e
d po
in
ts,
determ
ines the
“length”
of se
gment
1
1
IN
L
Y
Y
or the m
a
xim
u
m
of efficiency
M
P
K
of a two-po
rt ci
rcu
it:
M
P
IN
L
K
th
th
Y
Y
m
1
1
)
(
1
1
Thu
s
, on
e m
o
re fo
und
ed
invarian
t is eq
u
a
l
to
a co
n
c
rete n
u
m
b
e
r. Th
at
in
v
a
rian
t is si
milar to
th
e
cr
o
s
s-
ratio
o
f
t
h
e
h
u
m
an
body eq
u
a
l t
o
1
.
3
1
.
Fi
gu
re
1
1
.
a):
Tw
o-
po
rt
net
w
or
k ;
b):
i
n
p
u
t
-
out
put
c
h
a
r
act
eri
s
t
i
c
;
and
c):
m
ovem
e
nt
of t
h
e se
gm
ent
fo
r
d
i
fferen
t
in
itial v
a
lu
es
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
On
t
h
e
An
al
o
g
y
of
N
o
n
-
E
u
cl
i
d
ea
n
Ge
omet
ry
of
Hu
m
a
n
B
o
d
y
Wi
t
h
El
ect
ri
cal
N
e
t
w
orks (
A
sho
k
V
a
se
asht
a)
38
6
C
a
sca
d
e co
nnect
io
n of
t
w
o-po
rt
s
- Let
u
s
co
nsid
er th
e cascad
ed
two-ports TP1
and
TP2
i
n
Figu
re
12
. The rel
a
t
i
o
nshi
p o
f
regi
m
e
param
e
t
e
rs at di
ffer
e
nt
pa
rt
s
of t
h
e net
w
o
r
k
or “m
ovem
e
nt
” on t
h
ese pa
rt
s al
so
co
rresp
ond
s to p
r
oj
ectiv
e tran
sfo
r
m
a
tio
n
s
.
Th
e lo
ad
ch
ang
e
fro
m
th
e v
a
lu
e
1
2
L
Y
to
th
e
v
a
lu
e
2
2
L
Y
defi
ne
s
the corres
p
ond changes
1
L
Y
,
1
IN
Y
. The len
g
t
h
of seg
m
en
ts o
f
all th
e lo
ad
lin
es i
s
d
i
fferen
t fo
r
th
e u
s
u
a
ll
y
use
d
E
u
cl
i
d
ea
n
ge
om
et
ry
.
If the m
a
pping is viewed as the proj
ectiv
e t
r
an
sfo
r
m
a
tio
n
,
th
e in
v
a
rian
t,
wh
ich
is th
e cro
ss-ratio
of
fo
ur
p
o
i
n
t
s
, i
s
per
f
o
r
m
e
d and
de
fi
nes t
h
e sam
e
l
e
ngt
h
of se
gm
ent
s
. Th
us,
net
w
or
ks o
f
t
h
i
s
ki
n
d
ar
e
pr
o
j
ect
i
v
el
y
–
sim
i
l
a
r. There
f
ore
,
t
h
e
r
e i
s
s
o
m
e
ki
nd
of
t
h
e electro -
bi
ological
an
alog
y: d
i
spro
po
r
t
i
o
n
a
te
chan
ge
of se
g
m
ent
s
of l
o
ad l
i
ne f
o
r
di
f
f
ere
n
t
part
s o
f
a
net
w
o
r
k
co
rres
p
o
nds
t
o
gr
owt
h
chan
ges
of
pa
rt
s of t
h
e
h
u
m
an
bod
y.
Projec
ti
ve
pl
a
n
e
-
I
f
t
h
e
net
w
o
r
k
c
ont
ai
ni
n
g
t
w
o c
h
a
ngea
b
l
e
l
o
a
d
s,
p
r
o
j
e
c
t
i
v
e ge
om
et
ry f
o
r t
h
e
pl
ane ca
n
be
sh
ow
n.
I
n
t
h
at
ca
se, t
h
e
l
o
a
d
st
r
a
i
ght
l
i
n
es
f
o
r
m
t
h
e co
or
di
na
t
e
t
r
i
a
ngl
es
. T
h
eref
ore
,
net
w
or
ks
of
this ki
nd are
also
projectively - sim
ilar.
Fi
gu
re
1
2
. a
)
:
C
a
scade c
o
n
n
e
c
t
i
on
of
t
w
o t
w
o-
p
o
rt
s;
b):
C
o
rres
p
on
di
n
g
I-
V c
h
aract
eri
s
t
i
cs.
5.
CONFORMAL TRANSFORMATION
S OF
ELECT
R
IC NETWORK
Let u
s
con
s
id
er
a p
o
w
e
r
supply syste
m
w
ith
tw
o
vo
ltag
e
r
e
g
u
l
ator
s
2
1
,
VR
VR
and l
o
ads
2
1
,
R
R
sh
ow
n
in
Figu
re
1
3
. Th
e regu
lato
rs d
e
fin
e
vo
ltage
transm
ission coefficient
or
t
r
ansf
o
r
m
a
ti
on rat
i
o
2
1
,
n
n
. The
v
o
ltag
e
regu
lato
rs are
conn
ected
to
a li
m
ite
d
cap
acity supp
ly, vo
ltag
e
so
urce
0
U
.
An i
n
t
e
rfe
rence
o
f
th
e
regu
lato
rs on
reg
i
m
e
s o
r
lo
ad v
o
ltag
e
s
2
1
,
U
U
is observe
d
because
of e
x
iste
nce
of an inte
rnal re
sistance
i
R
.Let us
consi
d
e
r
the
case
2
1
R
R
R
i
.
Fig
u
re
13
.
Power su
pp
ly system
with
two
v
o
l
tag
e
regu
lato
rs
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 3,
J
u
ne 2
0
1
4
:
37
8 – 3
8
8
3
87
The net
w
o
r
k b
e
havi
or
or “
k
i
n
em
ati
c
s” vi
a vari
a
b
l
e
param
e
t
e
rs
2
1
,
n
n
i
s
descr
i
bed by
a sp
he
re i
n
t
h
e c
o
o
r
di
nat
e
s
i
U
U
U
,
,
2
1
in Figures
3
,
4
.
Fo
r regu
latio
n, it is
b
e
t
t
e
r
t
o
use suc
h
g
r
o
ups
of
t
r
an
sf
or
m
a
t
i
ons
or m
ovem
e
nt
s of
p
o
i
n
t
s
i
n
t
h
e pl
ane
s
2
1
,
U
U
and
2
1
,
n
n
, w
h
en
i
t
i
s
i
m
pos
si
bl
e t
o
de
duce
a
wo
r
k
i
n
g
poi
nt
ove
r t
h
e ci
rcl
e
s
,
w
h
i
c
h
co
rre
sp
on
d t
o
t
h
e
eq
ua
t
o
r
of
s
phe
re
b
y
a fi
ni
t
e
s
w
i
t
c
hi
n
g
num
ber.
In t
h
i
s
se
nse,
we de
ri
ve a hy
per
b
ol
i
c
geom
et
ry
. On t
h
e pl
ane
2
1
,
U
U
it
is th
e B
e
ltra
m
i-Klein
'
s
m
o
d
e
l
and on t
h
e
plane
2
1
,
n
n
it is th
e Poin
care's m
o
d
e
l. Th
e correspo
n
d
i
ng
circle carries th
e n
a
m
e
of th
e ab
so
lu
te
an
d d
e
fin
e
s an in
fin
itely remo
te bo
rd
er. Let
u
s
pu
t th
e
v
a
lu
e
0
2
n
. The
n
, t
h
e r
e
gi
m
e
change
goe
s o
n
l
y
o
n
axes
1
U
and
1
n
. Th
e
co
nfo
r
m
i
t
y
o
f
t
h
e ch
aracteristic po
in
ts and
run
n
i
n
g
po
in
t is sh
own
in Figure 14
.
Fi
gu
re
1
4
.
C
o
nf
orm
i
t
y
of t
h
e
va
ri
abl
e
s
1
1
,
n
U
o
f
t
h
e
hy
pe
rb
ol
i
c
t
r
ans
f
orm
a
t
i
ons
From
t
h
e m
e
tho
d
i
cal
poi
nt
of
vi
e
w
,
i
t
i
s
usef
ul
t
o
c
o
n
s
i
d
er
t
h
e
hy
per
b
ol
i
c
t
r
an
sf
orm
a
t
i
on
by
t
h
e
an
alog
y, co
rresp
ond
ing
t
o
the relativ
istic rule o
f
sp
ee
d c
o
m
posi
t
i
on i
n
r
e
l
a
t
i
v
e m
ovem
e
nt
m
echani
c
s.
If
, f
o
r
exam
ple
5
.
0
1
1
U
, the
n
5
.
0
2
1
U
and
i
t
do
es
not
de
pen
d
fr
om
the
val
u
e
21
1
U
.
In
the case
2
1
R
R
R
i
,th
e
sph
e
re
will b
e
tran
sformed
to
an
elli
p
s
o
i
d
.
Th
erefore, th
ese n
e
t
w
o
r
k
will b
e
con
f
o
r
m
a
lly
o
r
Mö
b
i
u
s
-
sim
ilar for the
plane
2
1
,
n
n
. The
r
efore, the
r
e is some kind of the
elect
ro - bi
ological analogy: the cha
nge
of
n
e
two
r
k
p
a
r
a
m
e
ter
s
cor
r
e
spond
s t
o
g
r
o
w
t
h
ch
ang
e
s of
b
i
o
l
o
g
i
cal
o
b
j
ects.
6.
CO
NCL
USI
O
N
Th
e an
alysis of
h
u
m
an
gr
owth
and
an
alysis of
o
p
e
r
a
ting r
e
g
i
m
e
s o
f
electr
i
c n
e
twor
ks show
s
an
inva
riant of
projective and c
o
nform
a
l transform
a
tions.
Di
ffe
r
ent types
of
the c
r
oss
-
ratio take
place for a
n
el
ect
ri
c
net
w
or
k. The
c
h
an
ge of
an
ope
rat
i
n
g regi
m
e
of
t
h
e gi
ven
net
w
o
r
k
or
c
h
an
ge
o
f
net
w
o
r
k
para
m
e
t
e
rs
resul
t
s
t
o
t
h
e
pr
o
j
ect
i
v
e
or
con
f
orm
a
l
sim
i
l
a
ri
t
y
of
net
w
or
ks.
T
h
e est
a
bl
i
s
he
d el
ect
r
o
-bi
o
l
o
gi
cal
an
al
ogy
devel
o
p
s
a m
e
tho
d
o
l
o
gi
cal
ba
si
s of
a
ppl
i
cat
i
o
n
of
no
n-
Eu
clid
ean
g
e
o
m
et
ries for these a
r
e
a
s.
REFERE
NC
ES
[1]
S.V. Petukhov
, “
N
on-Euclid
ean
g
e
ometri
es
and
algorithms of liv
in
g bodies,”
Comp
uters &
Mathematics
wi
th
Applica
tions
, 17, no. 4-6, pp. 505
-534,1989.
http
://www.sciencedir
ect.com
/science/article/pii/08981
22189902484
[2]
T.
Lundh, J. Ud
agawa, S
.
E.
H
a
nel,
and H. Otani,
“Cross- and tr
iple-
ratios of
h
u
man bod
y
par
t
s during
development,”
The Anatomical
Record
, vol.294, no.8
,
pp
.1360–1
369,2011.
[3]
Conformal geometr
y
, Availabl
e:
http:/
/en
.
wikiped
i
a.or
g
/
wiki/Conf
ormal_geometr
y
.
[4]
Stereograph
i
c pr
ojection, Availa
ble:
http
://
en.wi
k
ipedi
a
.org/w
ik
i/Stereogr
aphic_
projection.
[5]
H
y
perbol
ic
geo
m
etr
y
, Ava
ilab
l
e
:
http
://
en.wik
ip
edia
.org/wiki
/
H
y
perboli
c
_geom
et
r
y
.
[6]
A. Penin
,
“The invarian
t prop
erti
es of two-por
t c
i
rcuits,
”
In
ternational Journal of
electrica
l
and
co
mputer
engineering
, vol.4, no
.12, pp.740
- 746, 2009.
[7]
A. Penin
,
“Invar
i
ant properties o
f
cascaded
six-p
o
le n
e
tworks,”
I
n
ternational Jou
r
nal of
circuits,
systems and
signal processin
g
, vol.6, no.5
,
pp
.305-314, 2012.
Evaluation Warning : The document was created with Spire.PDF for Python.