Complex circular charts

COMPLEX CIRCULAR CHARTS BY

K. M O R I T A 1 A N D

Y. S E K I G U T I "°

ABSTRACT

This paper investigates the homographic solution of the functional relations of eight-complex functions, in which any one variable is unknown. The functional form treated here is t h a t t h e anharmonic ratio of the four-complex functions is equal to one of the remaining four-complex functions. The required chart is called a complex circular chart, and is solved nomographically w i t h t h e aid of both properties of circular transformation and conformal representation in the theory of functions of a complex variable. Special cases of the complex charts, including the complex concircular chart and t h e pseudo concircular chart, are studied and explained. 1. INTRODUCTION

Nomographic representation of the functional relations between two complex variables was studied by I. A. Vil'ner (1) 3 and F. Reutter (2), and one of the present authors and Y. Simokawa have found a new method of nomographic representation of the functional relations among three-complex variables (3). Furthermore, one of the authors has published two papers on the complex nomography of four- and fivecomplex variables (4), that are the extension of K. Ogura's real concircular chart (S). In this article the authors intend to show the nomographic solution of functional relations among eight-complex variables, which consist of the form (fl, f2, f3, f4) -- (fs, fe, fT, fs), where the parentheses denote the anharmonic ratios of each of the above four-complex functions. They call their chart the complex circular chart, and give some type equations and the corresponding charts as special cases. They also study the linear transformation of the chart to obtain a simpler form of it. 2. COMPLEX CIRCULAR CHART

Consider a square matrix of the fourth order of the form fl(Zl) ~/~/1"6 c

=

!l

[f,(zl)'fs(zs) 1

f'.'(Z2) f:~ (Z:*) f4(z4) f6 (z6) .fT(zT1 f8 (zs) , !1) f~_(z',.)'f6(z6) f3(z'~)@(ZT) f4(z4)'fs(zs) l

1

1

~Professor of Applied Mathematics, Faculty of Technology, Kanazawa University, Kanazawa, Japan. ~Leeturer of Mathematics, Department of General Education, Nagoya University, Nagoya, Japan. s The boldface numbers ill parentheses refer to the references appended to this paper. 49

50

K . MORITA AND Y. SEKIGUTI

[J. F.

I.

where zk = xk + iy,, i s = - 1, k = 1, 2, 3, • •., 7, 8, and t h e f ' s are analytic functions of the corresponding z's over an open region R. If, in the above matrix, four-column vectors (fl, fs, f l "f~, 1), - . . , (f4, fs, f4 "fs, 1) in the Euclidean space Re are linearly d e p e n d e n t , we have

det (Ms c) =

fl

f2

f3

f,

f5

f6

f7

f8

fl'f5

f2"fe

f3"f~

f,'f8

1

1

1

= 0,

(2)

1

a n d we call m a t r i x (1), with the above m e n t i o n e d condition of linear d e p e n d e n c y , a complex circular c h a r t matrix. T h e n (2) m a y be w r i t t e n in the form (f,, f~, f~, f,) = (fs, f~, fT, fs),

(3) 4

where the p a r e n t h e s e s denote the a n h a r m o n i c ratios of each of the fourcomplex functions. Rewriting (2), fl f3. f~ -- f3 = f5 -- fT. f8 -- f , f l - f , ' f 2 -- f4 f5 - f s ' f 8 - fs" -

-

(3a)

N o w consider first the n o m o g r a p h i c representation of the analytic function w = f ( z ) in a Gaussian plane. P u t t i n g w = f ( z ) = f ( x + iy) = u ( x , y) + iv(x, y), we get a p o i n t P ( z ) in the w-plane as an intersection of two curves, each belonging to t w o families of curves u = u (x, y), v = v (x, y), which have the indices of (x) and (y), respectively (Fig. 1).

(Y)

(/ 0

/~. 0"~

-~._

1't

iii._- ~o

FIG. 1. Nomographic representation of Z(z).

D

Fla. 2.

Nomographic solution of (2), the first method.

4 This functional equation was treated first by P. Luckey (6), in nomography of reals.

Jan., I96Z.]

COMPLEX CIRCULAR CHARTS

51

Then consider a linear transformation o~ in which the three distinct points Z1 {fl(z l) }, Z2 {f~ (z 3) }, Z, {fa (z,) } correspond to the three distinct points Zs{fs(zs)}, Zs{f6(z6)}, ZzIfT(zT)} in the w-plane, that is, a correspondence

fa f2 ffar. w =

fs

f6

(4)

Then Eq. 2 indicates that a point Z,{f4(z4)} corresponds to another point Zs{fs(z8)} by this linear transformation oa, shown by the expression

fl

f= f~ A f~f5 fff6 fff7 1

1

Z Z* =

Zg*

1

(5)

0.

1

Using this relation, we may obtain the following nomographic solution of functional equation (2) as a first method. We first plot the given seven points Zi, (j = 1, 2, - . . , 7), in the w-plane (Fig. 2), and let C,m, denote a circle, passing through three points Z,, Z~, Z., and let a,/3 denote the angles between Ca2a, Cla, and C12a, C234, respectively. Then the required eighth point Zs is, according to the property of circular transformation and conformal representation of linear transformation, an intersection point of two circles C57s and C6rs, which are making intersection angles a,/3 with them, respectively. As a second method, we use the relation

ZIZ8 Z2Z3

ZfiZ7 Z6Z7

Z , Z , : Z~Z, =

Z---~8 ZsZs"

(6)

Taking absolute values of both sides of (3a), so that

ZsZ8 . .

Z6Zs

ZlZ4 Z2Z3 Z5Z7 . .

c

ZIZa Z2Z4 Z6Z7

(6a)

( = known ratio from Fig. 3), and using the relation of amplitudes, we get z421z3

-

z42

z3 = z

( = known angle from Fig. 3). fore,

2

z7 -

z82oz7

=

0

Then, Z627Z5 -- Z62,Z5 = 0.

Z82sZ~ = Z627Z5 - -

0 =

,;.

(7)

There(7a)

52

K.

~{ORITA AND Y. S E K I 6 U T I

[J. 1;. I.

F r o m expressions (6a) and (Ta), we m a y obtain the required eighth point Z8 as one of the intersection points of an Apollonius' circle C with a known ratio c and a circle K with a known angle of circumference ¢, where K is to be taken at the same side of ZT, and the required intersection point Z8 be chosen between t w o intersection points of C and K b y the sign of angles (Fig. 3). Especially when 0 = 0, two pairs of four points 5 Z,, Z.,, Z3, Z4; Zs, Zs, ZT, Z8 are concircular, respectively, and our chart degenerates into the complex concircular chart of four-complex variables (4), having the t y p e equation ( f l , f2, fz~,f4) = a,

{

.

,

(a" real c o n s t . ) .

( 2~11

"T-.~ l

/,

" "6Z 3

iI / /

ff

Z,,

4~_ FIG. 3.

Nomographic solution of (2), the second method.

Example Figure 4 shows a nomographic solution of a functional relation

(Zl, z2, z3, z4) = (z5 ~, ze ~, zT, zs) b y the first method, where zi ( j = 1, 2, 3, 4, 7, 8) are indicated on the orthogonal rectilinear nets, and zs, z6 on the orthogonal parabolas. W h e n given values are zl = - 5 + i, z2 = - 3 + 3i, z.~ = - 1 + i, z4 = - 4 - 2i, z5 = 1.5 + 0.5i, ze = 2 + i, z7 = 5 + 2i, respectively, the required value z8 is 3.28 - 0.1i b y the constructions shown in Fig. 4. Since the exact solution is a p p r o x i m a t e l y 3.22 - 0.17i the relative errors, b o t h for the real and imaginary parts, are a p p r o x i m a t e l y 0.02. 5 We call the figure for {z,, z.2, z3, z,} the lirst partial chart, and for Iz~, z6, zT, z,} the second partial chart.

Jan., 1962.]

COMPLEX 2

4"

J

,,~

CIRCULAR 4"

CHARTS

53

,"

¢

'~

Flil

g

Io~

Y

0

tJ~

,c • i 0 4

i-o ~ o l

x.tol

trio,. 4.

y~

Nomographlc solution of a functional relation (zl, z.,, z.~, 14) = (152, 16~, 17, zs) by t h e first method.

(x~)

l"tc,. 3.

txj)

N o m o g r a p h i c sohltion of a functional relation (10) or (10a).

54

K. MORITA AND Y. SEKIGUTI a. VARIOUS s r s c r ~

[J. F. I.

TYPE S O U A T m r , S o r T r r s COMrL~-X CXRCUrAR CHARTS

(a) A Type Equation of Five Variables In the general type equation of the complex circular charts (see Eq. 3), putting f5 -~- ~ ,f6 ~ 0,f7 ~ 1,fs ~-f(z) and otherf's remaining invariant, we have (fl, f2, fa, f,) = f, (9) the type equation of the complex pseudo concircular chart (4).

(b) A Type Equation of Six Variables In this type equation

(fl, f2, fa, ~) = (f~, fs, fT, ~)

(10)

or fl

f2

fa

f5 f8 fr 1

(lOa)

= O.

1

1

When f4 = oo, fs = ~ in (3), that is, the point at infinity is a double point of homography, (10) reduces to (10a), and this type equation represents a complex chart of six variables shown in Fig. 5, in which z~Z1Z~Za ~ Zgg6Z7 by a principle of the elementary theory of functions of a complex variable, and the unknown point ZT(zr) is obtained by this similitude relation (Fig. 5), (7).

(c) A Type Equation of Seven Variables In this type equation,

(fl, f2, fa, f4) = (fs, f6, fT, ~)

(11)

or fl

-- fa.f2

-- f3

fl-A'f2-A

__ f 7

-- f,

fT-f6"

(11a)

Expressions (6) and (7) become, respectively,

ZTZ5

Z1Za Z~Za =

Z7Z6

__

-

c

(12)

Z1Z4"Z2Z4

and Z4~lZ3

-- Z4~,2Z3 :

Zg~7Z5 ~- O,

(13)

where z7 is assumed unknown. Then we may obtain the required seventh point ZT, in the second method, as one of the intersection points of an Apollonius' circle C with a given ratio c and a circle K, having a circumference angle 0 (Fig. 6).

Jan., I962.]

COMPLEX

CIRCULAR

CHARTS

55

b~

~

Y

0 t : m . 6.

fz. N o m o g r a p h i c s o l u t i o n of a f u n c t i o n a l r e l a t i o n (f], f~, .f:~, f 4 ) by the second method.

-z

Z

o

.

(f:. f,~ .i'; ~

~

\/

I

v

t".

4

=

;

\

I

""d :~ "2.

-2 Fro. 7.

0

z

-2

4

8

N o m o g r a p h l c s o l u t i o n of z~ -- z~ ." z2 -- z3 = z7 _ _-- zr, b y t h e s e c o n d met hod. ~1

--

Z4

"~2 - -

Z4

Z7 - -

Z6

56

K.

MORITA

AND V.

SEKIGUTI

[J. F. I.

We m a y also solve this functional relation by the first m e t h o d shown in Section 2. Example

Figure 7 shows a nomographic solution of (Zl, g2, Za, Z4) = (Z5, Z6, zr, ~ ), or Zl__.-- za z= - - za _ zr Z~ - - Z4 Z2 ~

Z4

z~, by the second m e t h o d , where all

Z7 - - Z6

z's are indicated on the orthogonal rectilinear nets. W h e n we have z~ = - 2 + i , z~ = 0.5 + 1.5i, za = 1 - i, z, = - 0.5 + i , z~ = 1 + 3 i , z~ = 5 + 4 i , respectively, the required value z7 is o b t a i n e d as 2.78 + 4.98i, whereas a value by the numerical calculation is 2.72 + 4.96i, a p p r o x i m a t e l y . 4. LINEAR TRANSFORMATION OF THE COMPLEX CIRCULAR CHARTS

In order to obtain a n e a t e r form of a complex chart, it is preferable to operate a linear t r a n s f o r m a t i o n , given by w =

a 1 1 z --~ a l 2

,

(14)

al,a..,.,. - - a 2 , a ~ ~ 0

a ,21z -~- (l ~_,.,

to the original chart. Rewriting the above expression by letting Wl

=

allZl

~

al.2.Z,2_,

W2 =

a21zl

-[- a2,2z.2,

w

= wl/w~,

z

=

(15)

and

wl'=a.z,'+a..z,.,',

w'=w,'/w/, zt=z//z./,

(lSa)

we have w,w(

= a11a22zaz/ + alsa21zltz2 + a , l a 2 ~ z l z /

w/w2

= a~2a2~z,z2 r +

Wl"Wl t =

alla22zx'z2 -[- a n a 2 1 z l z ~ '

a l l a 1 2 z l z 2 t -2v a l l a ~ z l ' z 2

+

w2w2 p = a21a2..zlz( --I- a2~a~2z~'z2 +

ala 2 Z l Z l

+ al~a~2z2z2 t, + a~a~z2z2',

(16) I "~-

a12 2 g2z2 t,

a212 ZlZl ! "-~ a222 Z2Z2t.

Representing these in the form of m a t r i x multiplication, we get M*

(17)

= TM,

where ZlZ2 t

7.017.02!

gltg2 l

M = Wl,Wl ! W2W2 t

Jan., I96z.]

COMPLEX CIRCULAR CHARTS

57

and aria22 T =

a.j2(t22 ,

a~2a,21 a l j a 2 t

a12a21 axja,~,z

aua.2~

a12a:2

anal2

anal2

au 2

a122

a2~a22

a21a2e

a2~ 2

a27

(18)

M, M* being equivalent to the original complex chart matrix Ms c and transformed complex chart matrix, and T being the transformation matrix, respectively, as z = z , / z 2 , z' = z//z,,.', w = ¢Lf.'I/W2, w ! = 'z~Jif/w2 t. From (17) and (18) we have (let (M*) = (let ( T ) . d e t (M),

det (T) ~ 0;

hence det ( M * ) = 0 follows from det ( M ) = 0; therefore, a given complex circular chart is transformed, by the linear transformation (14), into another complex circular chart which, by a moderate selection of the coefficients aj~ in (14), is in a more easily solvable form. As a special case, when we wish to transform the first partial chart only, rewriting the linear transformation expression (14) as W, = all• l --~ a12z2,

w 2 = a21z 1 J[- (J22z2,

ez~ = ~ll/w2,

z = •[/z2,

(19)

z' = z,'/z~.',

(20)

and z/

= 1 . z ? + O.z~.',

z2' = O.zl' + 1.z~',

we arrive at the transformation matrix T~, by a procedure similar to t h a t used in the preceding case: an

0

0

a12

0

a2-_ a21

0

0

a12

all

0

a21

0

0

a22

;

T l

(21)

and when only the second partial clmrt is transformed, the transformation matrix Tr is similarly given by

~

22 0

Tr = ;12

a21

0

i

all

0

(~12

0 a.21

all 0

0(t22"

(22)

58

K. MORITA AND Y. SEKIGUTI

[J.

F. I.

Furthermore, it is easily seen that a transformation matrix for the complex concircular chart is also expressed by (21). REFERENCES

(1) I. A. VIL'NER, in various Russian journals, for example, Akad. Nauk SSSR, .In. Appl. Math. and Mech., Doklady Akad. Nauk SSSR, Uspehi Matem. Nauk, Mat. Sbornik ; Izdat. Moskov, Gos. Univ., Moscov, etc. (2) F. REUTTER, "Nomographische Darstellung von Funktionen einer komplexen Ver~inderlichen," Z. angew. Math. Mech., Vol. 36, pp. 1-3 (1956). (3) KATUHIKOMORITA AND YAKITI SIMOKAWA,"Nomographic Representation of the Functional Relations among Three Complex Variables," Z. angew. Math. Mech., Vol. 40, pp. 350-359 (1960). (4) KATUHIKO MORITA, "Complex Concircular Chart," (in Japanese), Kagaku (Science) (Japan), Vol. 29, pp. 148-149 (1959). KATUHIKO MORITA AND YAKITI SIMOKAWA,"Complex Pseudo Concircular Chart," (in Japanese), ibid., Vol. 30, pp. 207-208 (1960). (5) KINNOSUKE, OGURA, "Keisan-Zuhyo (Nomography)," Tokyo, Iwanami Shoten, 1940, pp. 149-155. (6) PAUL LUCKEY, "l~lber graphische Rechentafeln mlt einer frei beweglichen Leiter," Z. angew. Math. Mech., Vol. 7, pp. 155-158 (1927). (7) KATUHIKOMORITA AND YAKITI SIMOKAWA,"Nomographic Representation of the Functional Relations of Several Complex Variables," to be published in Z. angew. Math. Mech.