The T and CLP families of triply periodic minimal surfaces. Part 3. The properties and computation of CLP surfaces

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The T and CLP families of triply periodic minimal surfaces. Part 3. The properties and computation of

CLP surfaces

Djurdje Cvijović, Jacek Klinowski

To cite this version:

Djurdje Cvijović, Jacek Klinowski. The T and CLP families of triply periodic minimal surfaces. Part

3. The properties and computation of CLP surfaces. Journal de Physique I, EDP Sciences, 1992, 2

(12), pp.2207-2220. �10.1051/jp1:1992102�. �jpa-00246696�

Classification

Physics

Abstracts

02.40 61.30 68.00

The T and CLP families of triply periodic ^{minimal surfaces.}

Part 3. The properties and computation ^{of CLP surfaces}

Djurdje Cvijov16

and Jacek Klinowski

Department

of

Chemistry, University

of

Cambridge,

Lensfield Road,

Cambridge

CB2 IEW, G.B.

(Received J7 June J992, accepted ²⁴

August

J992)

Abstract. Parametric

equations

for normalized CLP minimal surfaces

provide

the solution of the

problem

of

finding

the coordinates of the CLP saddle surface inscribed in _a

given tetragonal parallelepiped (right tetragonal prism).

This is crucial for the

matching

of

specific

surfaces to real

structures. The geometry of _a CLP surface

depends only

on the ratio cla of the

tetragonal

_axes, and

can be described in terms of _a

single

free parameter. ^{We offer} a choice of two such parameters, both related _to surface geometry, ^{and derive}

analytical expressions

for their

relationships

to the

axes ratio and the norrnalization factor. Parametric equations ^for norrnalized CLP surfaces enable us, for the first time, to find the surface

corresponding

to any

given

value of the cla ratio.

Straightforward physical applications

are therefore

possible.

A CLP surface is

perfectly

self-

adjoint only

when the free parameter A =0. We list exact coordinates of CLP surfaces

corresponding

_to

prescribed

values of the cla ratio.

Introduction.

Previously

^of interest

only

_as mathematical

objects [1-3], triply periodic

embedded minimal surfaces

(TPEMS)

have been

applied

to real

problems

in many areas of

physical

and

biological

sciences

[4].

In

particular, they

are well suited for the

modelling

of condensed matter, as

suggested by

Scriven

[5], Mackay [6-7], Hyde

and Andersson

[8], Mackay

and Klinowski

[9]

and Sadoc and Charvolin

[10-11]. Rigorous

mathematical

description

of 20 surfaces in terms of the

Enneper-Weierstrass representation [12-17, 26]

will

undoubtedly

lead to further

applications.

The tD

(also

known _as

T)

and the CLP families _are the

only

TPEMS with known

parametric representations,

derived

by

_us in the first paper ^of the

present

^series

[18].

The

second paper

[19]

has

demonstrated,

for the first

time,

the

possibility

of

straightforward applications

of tD surfaces _as models

(without adjustable parameters)

for

matching

actual

structural data. We _now demonstrate that such

possibility

exists for CLP surfaces.

The CLP

family

of TPEMS.

The CLP surface _was first described

by

Schwarz

[I]

and named

by

Schoen

[2].

The _name refers to the

graph

of the tunnel network of the surface which is in the form of Crossed

Layers

2208 JOURNAL Dfi PHYSIQUE I N° 12

of Parallels. CLP surfaces have

tetragonal _symmetry,

and _can be

generated by using

two _non-

congruent

types ^{of skew}

straight-edged 6-gons only [20].

One of them is the

6-gon

obtained

by taking

six of the nine

edges

of _a

prism

^the bases of which _are

right-angled triangles (see Fig, la).

The solution of the Plateau Problem for such

6-gon

is _a finite minimal surface

piece

of the CLP surface

(see Fig. lb)

which _can be used _{as a}

building

block for

constructing

_an infinite

minimal surface

(Fig, lc).

We consider the

properties

of _a finite minimal surface

piece

of the CLP surface

(referred

to _as the CLP saddle

surface)

which _can be

thought

of _as inscribed in the

right tetragonal prism

with

edges

_{a, a} and _c

(see Fig, ld).

^{The CLP} saddle surface is bounded

by

four

straight-line segments (lying

on the

edges

of the

right tetragonal prism)

and four curvelinear

segments (lying

ⁱⁿ the bases of the

right tetragonal prism),

and contains two

straight-line

segments which divide it into four congruent parts.

We will show that CLP surfaces make up a one-parameter

family

of

triply periodic

embedded minimal

surfaces,

I.e. _can be described in the form

X#X(U, V(p) Y"Y(U,V(p) Z#Z(U, V(p)

where _u and _{v are}

parametric

variables with values in _a

specific parametric domain,

and p is a free parameter. ^{The semicolon} stresses the difference between

parametric

variables and the free parameter. ^{In other}

words,

for _a

particular

value of p we have

parametric equations

of _a

specific

surface. The free parameter ^is any

quantity

with _{a one-one}

correspondence

to the cla ratio of the

edges

of the

right tetragonal prism

in which the CLP saddle surface is inscribed.

This _means that different CLP surfaces

correspond

to different values of the _same value of the

free

parameter,

^{and that surfaces}

corresponding

to the _same value of the free

parameter (I.e.

with the _same cla

ratio)

are identical.

We will offer _a choice between two free parameters, ^{and show that both}

correspond

to the cla ratio _on the one-to-one basis. We will also demonstrate that surfaces with the _same cla ratio _can be

distinguished by introducing

the

multiplication

_constant,

dependent

_on the cla ratio, which is referred to as _« normalization factor _» and denoted

by

K. For different

values of _K

(corresponding

to different values of

lengths

a and _c but _to the _same

cla

ratio)

a

multiplicity

of surfaces are

produced,

and the coordinates of these differ

by

the

multiplication

constant _K.

The free

parameter

^{E with 0} _~ ^E _~

l12,

_determines coordinates of the

images

of flat

points

on the unit

sphere

(± E,

±

/$, 0)

and

(± fi,

± E, 0

which _are obtained from the Gauss map of _a

specific

CLP saddle surface. The free parameter

A

(with

2

~ A

~

2)

_appears in the Weierstrass function of CLP saddle surfaces

[12]

R(r)

=

~

~

Since the parameters ^{A and E} are related

A

=

16(E~ -E~)

₂

E=

~

either of them _can be used.

(a)

~~

(b)

c ⁱ

i i

' ,"

,

' ''

' '

'

a

(c) (d)

a

Fig.

I. (a) Skew space

6-gon

(bold lines) made up of

edges

of _a tetragonal

prism

based _{on a}

right- angled triangle

_; (b) CLP minimal surface

piece spanning

(a) _; (c)

Arrangement

of

prisms leading

to an

infinite CLP surface (d) CLP saddle surface obtained from (c)

by dissecting

the

prisms

with

parallel planes

at

height

c/2 and c/2.

The

parametric representation

of the

family

of CLP saddle surfaces

(see Appendix

and Ref.

[18])

involves _a

special

function known as the

incomplete elliptic integral

of the first kind F

(q~, k),

where _{we assume} that the modulus k is real and lies in the interval

[0, 1]

and the

amplitude

_q~ is _a

complex

number. In what follows _we often _use the

properties

of

2210 JOURNAL DE

PHYSIQUE

I N° 12

z

Fig. 2.

-Right-handed

Cartesian coordinate system ^attached to the CLP saddle surface described by

equations

(1).

F

(q~,

k

)

which _were outlined earlier

[18-19].

^Detailed

theory

of

elliptic

functions _can be found in standard texts

[21-22].

The Cartesian coordinates _x, y and _z of a

specific

CLP saddle surface _are

given by

the

following parametric equations

x(u, v)

=

Kg~Re [F(arcsin ll'~(w ), k~)]

2

z(u,

_v

)

=

j

Kg~

K(k~)

_{Kg~ Re}

iF (arctan

~l~~(w

), k~)j

where Re is the real part ^{of the}

complex

F (q~,

k)

and

1l'~(w)= ~~°

gxw +1

~'~~~

l ₊

w~

are the

complex

functions where

w is

complex

_w

= u + iv, with _{w w} I. The

multiplication

factors g~ ^and g~ and the moduli k~ and k~ are defined in table

I,

and K stands for the

complete elliptic integral

of the first

kind, K(k)

=

F

(w/2, k).

All components ^of

equations (I) depend

on the value of the _same free real

parameter,

^{and such}

dependence,

except ^{for the}

normalization factor _K, is

given

in table1.

Table I.

-Multiplication

constants g~ and g~ ^{and moduli} k~ and k~ used in

parametric equations for

the CLP

surfaces expressed

in terms

of free

_parameters E and A. The domains

of

E and A _are

(0, l12)

_and

_{(- 2, 2),} respectively.

/

₂

~~

~/l

+

~/1 4(E~ E~) ~

l 2

~~

l+2

E~ 2+fi

~

ⁱ

-Vi _-4(E2-E4) 2-Wm

~

l+~/1-4(E~-E~) ^2+fi

~

⁴

Vi

₄

_{(E2 E4)}

8

Wm

~

[l

+

~/1 4(E~ E~)]~ ^[2

+

fil~

Quantitative

characteristics of the CLP

family.

An

analytical expression

for the cla ratio

(which

_we

designate by x)

for the CLP surfaces

cannot be derived from the

Enneper-Weierstrass representation

alone. The

parametric

equations (

I

)

show

that,

for any value of free parameter, ^the

point

A in the

complex plane

with coordinates

(1, 0)

is

mapped

into

point

^A on the CLP saddle surface with coordinates

(x,

_y,

c/2) (see Fig. 3a). Similarly,

the

point ( l12, l12)

_{in the}

_{complex plane} corresponds

to the

point (a/2, a/2, 0)

on the saddle surface

(see point

A in

Fig. 3b). Thus,

from

(I)

_we obtain

a = Kg~ Re F arcsin

~,

k~

g

x ~~~

c = Kg~

K(k~)

Since for any value of the free parameter we have

~~~"~

j~ ⁱ

_~

~~'~~~~

where

fi(g~)

is _some function of the free

parameter

^alone

(its

exact

analytical

form is

irrelevant),

the

elliptic integral

in

(2)

_can be

expressed

in the form

F

)

₊

_ifi(g~),

k~)

⁼

^K(kx)

⁺

^{if2(gx) (3)}

This makes it

possible

_to

_separate

the real and the

imaginary parts.

^From

(2)

and

(3)

we have

~ ~~~

~~~~~

(4)

c ₌ Kgz

K(kz)

and

gz

K(k~

'~ ~ ~

gx

K(k~)

2212 JOURNAL DE PHYSIQUE I N° 12

(a)

_jmjw)

A c

~

Re la)

~ ^D

(b) Im(w)

B A

Re _(W) c

c D

Fig.

3. Construction of characteristic parts ^{of the CLP surfaces.}

Using

table I, the ratio

(5)

_can be

easily expressed

ⁱⁿ terms of either of the parameters E _or A, _so that

expressions

for _x

= x

(E)

and _x

= x

(A

are

readily

available. We _see therefore that the

tetragonal

axis ratio _x is

independent

of the normalization

factor,

and

only depends (in

a continuous

manner)

_on the free

parameter

on the one-to-one basis. It follows that

x is

uniquely

characteristic

(an invariant)

of _a

specific

CLP surface.

It is

important

to determine the coordinates of _a surface for any

prescribed

value of xo. To do that, it is _not necessary ^to find

explicit analytical expressions

E

=

E(x)

and

A

=

A

(x), especially

since such

expressions

are

complicated

and involve Jacobi's

elliptic

functions. It is sufficient to solve the

equation

g~K(k~)

g~K(k~)

^{~° ~~~}

in _terms of either of the parameters ^E or A

by using

^{table I.} The

equation

is transcendental, and

can be solved

only numerically. Equation (6)

can be solved

directly by

^{numerical methods}

altematively

an

approximate equation

_can be solved.

To solve

(6) directly,

_we

adopt

the

following procedure.

To shorten the

root-searching

interval, _{we express x} in terms of E. Since there is

always only

one real _root

(I.e.

E)

for any value of xo and the interval is

known,

it is _not necessary to

employ root-finding

methods which involve derivatives. This is fortunate,

considering

the very

complicated expression

for the derivative of

(5).

To solve

(6)

_we have used the Van

Wijngaarden-Dekker-

Brent

algorithm [23].

In most

applications

^{it is} sufficient to

approximate

_x

by

a rational function. We _{use a} rational

approximation

of the form

£

4 _{p~ E'}

X ~

~

~

(7)

£

_q~ E'

0

where E is the free parameter ^{with 0} ~ E

~

l12,

_{and the} coefficients p~, q~

(I

₌

0, 1, 2,

3 and

4)

need to be determined.

By employing

the standard method of rational

interpolation [23]

_we have found rational

approximations

for _x

(see

Tab.

III)

with _a numerical accuracy better than 7

significant figures. Thus,

for _a

given

_xo, the determination of free parameter ^{is reduced} to numerical solution of the

quartic

£

4 _{~Pi xo}

qi)

E~

= 0

,=o

with coefficients

given

in table III.

Unfortunately,

it is very difficult _to

approximate

X for very

large

and very small values of the free parameter, so there is a need for _an

asymptotic expansion

of _some kind.

Table II.

Limiting

^values

of

the various constants

for

E

- 0

(corresponding

to A _-

2)

and

for

E

-

l12 (corresponding

to A

-

2).

~x ~z k k X K

E-o I 0 _~x~ 2/w

E -

l12 / _1/2

_{0 0 0}

Table III. Three rational

approximations

in

form (7) for

the ratio

x(E) for different

intervals

of

the

free

_parameter E.

0.06£ E < 0.20 o.40 SE £0.63

0

1 lls.9511514706845 820349143156813

2 %73 985950387064 -75.~77308

3 152.7952883144366 103.3426687204758

4 17178 77852003997 -

2214 JOURNAL DE PHYSIQUE I N° 12

The

expressions (4) readily yield

the

following

relations for the normalization factor

K for CLP saddle surfaces

K = a

(8a)

gx ^K

(k~)

K = c.

(8b)

gz

K(k~ )

Equations (8a)

and

(8b)

and

expressions

in table I

give analytical

formulae for the normaliz- ation factor

K in terms of the free parameter. ^{It is clear that for}

given

values of

a and _c,

K is

uniquely

determined _so that _can be used in either

(8a)

_or

(8b).

^{We will} use

(8a) exclusively.

For _a

specific

value of the

length

of the

edge

_{a, K}

depends only

_on the value of the free parameter ^and

changes continually

with

it,

with

limiting

values listed in table II. The

family

of CLP saddle surfaces normalized

by (8a)

are related _to

right tetragonal prisms

with the

same base

edge

_a and different

edges

_c. It is therefore

possible

to compare some characteristics for different surfaces

belonging

to the CLP

family.

Parametric

equations

for normalized CLP surfaces.

According

to

(1), (5)

and

(8a),

for any

specific

values of the

length

of

edges

_a and

c

(where

_{we assume} that _a is the

edge

of square basis and _{c as} the

height

of the

right tetragonal prism)

there is

always exactly

_one CLP saddle surface

completely

described

by

the

following

parametric representation

_:

y

(r,

b

=

-

^/ _Re

_{iF (arcsin ~l~~}

_(I

_w

where k~, k~,

ll'~(w )

and

ll'~(w )

all

depend

on the value of the free parameter, ^{have the} same

meanings

_as in

(I)

and _w

= r cos 0 ₊ ir sin 0 with 0 _{« r «} I and 0 _~ 0 _« 2 _w.

Unfortunately,

there is _{no true} addition formula for the

elliptic integral

F

(q~, k)

with _a

complex amplitude,

_so

that F

(u

+ iv,

k)

cannot be

expressed

in terms of F

(u, k)

and F

(iv, k) only.

It is therefore

impossible

_{to separate} the real and

imaginary

parts, ^and

parametric equations (9)

cannot be

simplified

further.

Parametric

equations (I)

and

(9)

_were derived with the

assumption

-2~A~2

(0

~E~

l12),

_I,e,

_assuming

_{finite values of}

a and _c. Two

limiting

cases need to be considered _:

(I)

_a is finite and _c infinite and

(it)

a is infinite and _c is finite. It is clear that then

equations (9)

cannot be

applied,

but it is easy ^to

verify numerically

that for A

= 2 and

cla - ~x~

equations (I)

with _K

=

2

give

the well-known

doubly periodic

Scherk's minimal

surface

(«Scherk's

first surface» in Nitsche's

terminology)

z=In

(cos y/cosx) [24].

Similarly,

A

= 2 and cla

- 0

corresponds

to Scherk's

singly periodic

tower of order _one

(which

Nitsche calls _« Scherk's fifth surface

»),

which also _can be

easily

described in terms of

elementary

functions _as sin _z

=

sinhx sinh _y

[24].

Parametric

equations (I)

_can therefore be extended to the these

limiting

cases.

It is

interesting

to use our

parametric equations

to

verify

Schoen's claim

[2]

that _a

specific

CLP surface is

self-adjoint.

Let S be the CLP saddle surface described

by

the

parametric

representation (9)

for

particular

^{values of} a and _c.

Then,

the surface S* with the Cartesian coordinates

z*(r,

0

=

~ Re

iif (arctan

_~l~~(w

), k~)j

is

adjoint

to S. It may seem at first

sight

that S* is rotated in

respect

to the chosen coordinate system ^{shown in the}

figure 2,

and that _a

straightforward comparison

is _not

possible. However,

the

problem

is

easily

resolved

by

the

following

transformation of the coordinates

which,

of

course, cannot affect the surface itself

X*

=

#(X*+y*) Y*=#(X*-y*)

z*~_~*

and

gives

identical coordinate systems ^for S and S*. The conclusions of detailed

analysis

are as

follows _:

I. For the surface S with A

=

2,

its

adjoint

surface S* is identical to the surface with the coordinates determinated

by (9)

and A ₌ 2. This confirms that Scherk's fifth and first surfaces _are

adjoint

to _one another.

2. For any 2

~ A _~ 2 with the

exception

of A

=

0,

the

adjoint

surface S* is

always

_a

member of CLP

family,

but S and S* _{are not} identical. This _can be illustrated

by comparing

their cla ratios shown in

figure

4.

4

~

3

W Xs

~

~

x~*

0

0 2

Free

parameter

^~

Fig.

4.

Comparison

of the cla ratios, XS and XS* for the CLP surfaces and for their

adjoint

surfaces.

2216 JOURNAL DE

PHYSIQUE

I N° 12

3. A CLP surface is

perfectly self-adjoint only

when A

= 0. This _means that the minimal

surface with the Weierstrass function

~~~~

"

fi

is

completely unique.

Computation

^{of CLP} surfaces.

The main obstacle _{to a} wider

application

of TPEMS in

experimental

science is that _most of

them have been described

empirically,

without the

precise

mathematical

specification

necessary for a

quantitative comparison

with

physical systems.

^{It is therefore essential} to

quantify TPEMS,

and _to establish

straightforward procedures

to compare such surfaces with actual structures. Even in the _case of

mathematically

well-described surfaces

(with

known

Enneper-Weierstrass representations),

^such

procedures,

as well _as

quantitative computation,

are

missing.

Among published drawings

of CLP saddle surfaces

only

that obtained

by

Terrones

[25]

and based _on the

Enneper-Weierstrass representation

can be

regarded

_as

quantitative.

In

general,

it is very difficult to

apply

the

Enneper-Weierstrass representation

to match actual structural data to a CLP

surface, mainly because,

without the

knowledge

of _K and _x, the

Enneper-Weierstrass representation

must be

regarded

_as a model with

adjustable

parameters. ^{In other}

words,

it does not tell _us how to generate a surface which would match

experimental

data. Parametric

equations

for normalized CLP surfaces

(9) provide

^the solution of the

problem,

which _can be

geometrically

formulated _as follows _: find the coordinates od the CLP saddle surface

(see Fig. ld)

which is inscribed in the

given right tetragonal prism.

A _more

practical

restatement of the

problem

is _: match the

given

structural data

belonging

to

tetragonal

_symmetry and confined to the volume of _a

right tetragonal prism

to _a definite CLP surface. This _means that _{we must :}

(a)

find the value of the free

parameter

^for

given

_c and _a, I,e. solve

equation (6)

with xo =

cla. The free parameter

fully

determines _a

unique

CLP saddle surface

(b) apply (9)

and calculate the coordinates for this value of the free parameter;

(c)

establish the

agreement

^{between structural data and the} coordinates

ofpoints

_on the CLP surface to a

prescribed degree

of accuracy.

Table IV. Constants used in the

computation of

the various CLP saddle minimal

surfaces for different

values

of

the cla aXis ratio _x.

=2 =0.5

R 1.36726376 0.098205240 1.52487530

K 0.6165553 0.55652108 0.44477396

1.02128444 1.088008572 1.21958182

o.71544904 o.s799s923 o.sis8oo9

0.20741721 0A2867546 0.69812594

kz ^0.99907413 0.98297064 0.87319008

This

procedure,

for the first

time,

enables easy and

straightforward practical applications (minimal

surfaces appear in _a

variety

^of

physical problems)

and

completely

obviates the need of

using

the

Enneper-Weierstrass representation.

This is not to say that the _use of the Weierstrass function, _necessary for the calculation of the Gaussian curvature, metric and

topology

_can be avoided. However, our

parametric equations completely

avoid the

integration

of the Weierstrass function.

Equations (9)

contain _no

adjustable parameters

^{and allow} us to use

« yes-or-no »

modelling.

This _means that it is

always possible

to say whether _{or not}

given

data match _a

specific

CLP surface _{to a}

prescribed degree

of accuracy.

Table V. Cartesian coordinates

of

three

different

CLP saddle

surfaces.

The columns

differ

in n~ and the _rows in no.

a)

_x

= 2.0.

b)

_x

=

1.0.

c)

_{x =} 0.5.

1 2 3 4 S 6 7 8

o.isioo63 o

1

0.071048284

2 0.029%6423

3

30089514 4

a)

1 2 3 4 S 6 7 8

0.35844771

o o

1

0.40913367

2 0.055915786 0.3075348

0.024101157 0.l1625311 3

049446296 4

b)

2218 JOURNAL DE PHYSIQUE I N° 12

Table V

(continued).

1 2 3 4 5 6 7 8

o

i o.036417933

2

3

4

C)

Part

(c)

of the above

procedure

will be described elsewhere

using rigorous

mathematical

model

theorey.

^Here we use parts

(a)

and

(b)

to compute ^{CLP saddle surfaces} for

x =

2,

and 0.5. The

length

of

edge

a is assumed to be I

(c

= x

)

and all

quantifies required

in the calculation _are

given

in table IV.

Computational

evaluation F

(w, k)

and

K(k)

has been

described earlier

[19].

Tables Va-c list

only

the coordinates of

1/8

of each saddle surface.

These surface

pieces

_are

asymmetric units,

and for each the

computational

domain

(r,

o

)

is _{a sector} of the unit disc with 0 _{w r w} I and 0 _w o _w arm. The sector is divided

using

a

8 _x 5

grid,

where the coordinates of

grid points

are

given by

w

(n~, n~)

_{= n~ ro} cos

(n~ oo)

₊

in~

_{ro sin}

(n~ oo)

with ro =

1/8 and

oo

=

ar/16

(n~

₌

1,

8 and n~ =

0, 1, 2, 3, 4).

It is clear that the coordinates of _a

complete

CLP saddle surface _can be obtained

by symmetry

considerations alone. Coordinates of surfaces

corresponding

to the _same value of _x but to a different value of

a are obtained

by multiplying

the coordinates in tables Va-c

by

_a.

Appendix.

For the

parametric equations

of tire CLP saddle surfaces

(Eqs. (9), (10)

and Tab. IV with 2 ~ A

~ 2 in Ref.

[18])

_to be

practically useful,

it is necessary to introduce the normalization factor _w, examine the

complex

square root function of the

complex

variable which appears in

equations

for the coordinates _x and y, and find _a suitable coordinate system. ^{It is}

clearly

desirable _to choose the _same branch of

complex

_square root for both coordinates _on the entire unit disc. We consider

only

that branch of the square ^root which is I at I. In the

equation

x ₌ xo ⁺ wgx Re

IF (w~, k~)j

where g~ and k~ are

given

ⁱⁿ table I, ^the

amplitude

_w~ defined

by

~ 2

w

~

Sl~ ~x

g~ w 2 +

can be

simply expressed

in _terms of the inverse sine function

only

p~ =

arcsin ^°~

gx _w +

and then the _x coordinate

correctly represents

^{the CLP saddle surface} on the entire unit disc.

However,

in the

equation

y = yo + Ngx Re

jif (w~,

k~

)j

with k~ =

[the _expression

_for

_amplitude

p~ ^is

(~+fi)w~

sin~wy~

4~~2

_2-A+1

which _means that the square root function cannot be avoided. With the above

assumption

about the branch of the square root, ^the

equation

for _{the y} coordinate

gives

_{a correct}

representation

of the CLP saddle surface in the first and the fourth

quadrant

of the

complex plane. Symmetry

considerations

require

that for the y coordinate the surface be

correctly represented

_on the upper

(I

and II

quadrant)

_part of the unit disc. It is therefore sufficient to

change

the

sign

of the

equation

in order _to represent ^{the lower}

part

^of the unit disc

correctly.

To resolve this

problem

we must «rotate» the solution function for the y coordinate counter-clockwise

by

ar/2. The

required

function for y is found

by using

the

procedure

described in

[18].

The _new

equation

for the y coordinate is

= Yo Ngx Re IF

(w~, k~)]

where g~ ^and k~ are defined in table I and the

amplitude

_w~ ^is

2 wi

w~ = arcsm

gx I ~ w

It is clear that the

relationship

between _p~ and p~ ^is

simply p~(w )

=

p~(iw ).

Finally,

for xo =

0,

_yo

= 0 and zo = wg~

K(k~),

we obtain the

right-handed

Cartesian 2

coordinate

system

^{shown in}

figure

2.

References

[i] SCHWARz H. A., Gesammeite Mathematische

Abhandlungen (Veriag

Julius

Springer,

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(1985)

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