The oCLP family of triply periodic minimal surfaces

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The oCLP family of triply periodic minimal surfaces

Djurdje Cvijović, Jacek Klinowski

To cite this version:

Djurdje Cvijović, Jacek Klinowski. The oCLP family of triply periodic minimal surfaces. Journal de

Physique I, EDP Sciences, 1993, 3 (4), pp.909-924. �10.1051/jp1:1993172�. �jpa-00246772�

Classification

Physics

Abstracts 01.55

The OCLP family of triply periodic minimal surfaces

Djurdje Cvijov16

and Jacek Klinowski

Department

of

Chemistry, University

of

Cambridge,

Lensfield Road,

Cambridge

CB2 IEW, U.K.

(Received 12 October 1992,

accepted

in

final form

20December 1992)

Abstracto- CLP surfaces with orthorhombic distortion (OCLP for short) are a

fattily

of two- parameter

triply periodic

embedded minimal surfaces. We show that they

correspond

to the

Weierstrass function of the form ^~ where

N/T~

_{(A + B)} T~ ₊ (2 ₊ AB) ^T~ (A + B) ^T~ + I A and B _are free parameters ^{with 2} _< A, B

~ 2 and A

> B, _T is

complex

with _{T «} I and

K real and

depends

_on A and B. When B

= A, the OCLP

family

reduces _to the one-parameter CLP

famfly

with

tetragonal

_symmetry. The

Enneper-Weierstrass representation

of OCLP surfaces

involves

pseudo-hyperelliptic integrals

which _can be reduced _to

elliptic integrals.

We derive

parametric

equations ^for OCLP surfaces in _terms of

incomplete elliptic integrals

F (#, k) alone.

These

equations completely

avoid

integration

of the Weierstrass function, thus

making

the _use of the

Enneper-Weierstrass representation

unnecessary in the

computation

of

specific

OCLP surfaces.

We derive

analytical expressions

for the normalization factor and the

edge-to-length

ratios in terms of the free parameters. ^{This solves the}

problem

of

finding

the OCLP saddle surface inscribed in

given

_a

right

tetragonal

prism,

crucial for the modelling of structural data

using

_a

specific

surface, and enables

straightforward physical applications.

We have

computed exactly

the -coordinates of OCLP surfaces

corresponding

to several

prescribed

^{values of the}

edge-to-length

ratio.

Introduction.

A

triply periodic

embedded minimal surface

(TPEMS)

is _an surface in three-dimensional space which is

infinite, minimal, periodic

in three

independent

directions

(I,e.

has space group

symmetry)

and free of self-intersections

(embedded). Approximately

40 TPEMS have been described

by

various methods

(now mainly crystallographic) [1-4]. Previously thought

of

only

as mathematical

objects [5],

in the last 20 years TPEMS have become of

great

^interest to

physicists, chemists, biologists

and material scientists

[6-7].

Of

primary

interest is the fact that these surfaces possess translation symmetry ^{which in}

principle

enables them to be matched to actual structures.

Also,

since _a TPEMS divides three-dimensional space into _two

disjunct regions (labyrinths)

in such way that each

region

is

multiply

connected and the _structure

bicontinuous, descriptions

of

interpenetrating crystalline

structures with

large

unit cells and

complicated

networks of cages ^{and channels} are

possible.

TPEMS _{are a} useful

crystallographic

910 JOURNAL DE PHYSIQUE ^{I N° 4}

concept

^{for the}

description

of condensed _matter, and _are

increasingly frequently

^referred to as

crystallographic

^minimal surfaces

[8-13].

In

general,

the local

Enneper-Weierstrass representation [5]

w

x ₌ Re

(I r~) _R(r)

dr

w~

y = Re

l~ ^I(I

⁺

^r~)R(r)

^dr

^(I)

w~

w

z ₌ Re 2

rR(r )

^dr

w~

where

i~

=

I and _r

= r~ +

ir~,

enables us to associate with every function

R(r) (the

Weierstrass

function), analytical

in _some

simple

connected

region

of

e, except

at isolated

points,

_a

unique parametrized

surface

r(r~, r~)

which is

guaranteed

to be _a minimal surface.

However,

such _a surface is _not

necessarily

free of self-intersections and its

possible periodicity

must be examined

separately.

The Cartesian coordinates of any

point

are

expressed

as the real

part (Re)

of contour

integrals,

evaluated in the

complex plane

from _some fixed

point

wo to a variable

point

_w. A

specific

minimal surface _can be determined

by integrating

^its

Weierstrass function. So

far,

the

integrals (I)

have been evaluated

analytically only

for _a handful of minimal

surfaces,

such _as

Enneper's

and Scherk's surfaces

[5].

However,

they

_can

((ways

_be evaluated

by

numerical

integration.

The crucial

problem

in

theory

of TPEMS is

deriving

the Weierstrass

function,

and since _a

new method for its construction has been

developed [14-17], approximately

^{20 different}

surfaces have been described in terms of the

Enneper-Weierstrass representation. However,

it has been shown that

one-parameter

^tD

(and

its

adjoint tP) [18-19]

and CLP

[18, 20]

families of

TPEMS,

_as well _as

zero-parameter D,

G and P surfaces

[21]

_can ^be

parametrized

ⁱⁿ terms of the

special

^{function known} as the

incomplete elliptic integral

of the first kind.

Rigourous

mathematical

description

of these surfaces

completely

avoids the

integration

of the Weierstrass

function,

and their

computation

obviates the need for

using

the

Enneper-Weierstrass representation.

We _now demonstrate that his is also true for OCLP 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

quantative 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

a known

Enneper-Weierstrass representation),

such

procedures,

_as well _as

quantative computation,

_are

missing.

This

work,

_as well _as

previous

_papers

[18-20]

stresses the

computational

aspects ^of the

theory

of TPEMS.

The

Enneper-Weierstrass representationo

We do _not know how many different minimal surfaces _can be described

by

the Weierstrass function of the form

~~~~ ~/r~

+ 2

pr~ +~ _r4

+ 2

pr~

₊ ^~~~

where _r

= r~ + I r~, while the normalization factor _K and the coefficients _p and A _are real.

These surfaces _can be

non-periodic,

_or

singly, doubly

_or

triply periodic. Further, they

_can be

embedded _or unembedded. The Weierstrass function

(2)

is

single-valued

_on the two-sheeted

compact

Riemann surface of genus 3 with

eight

branch

points

of order _one, and their

distribution makes it

possible

to

classify

all minimal surfaces described

by (2).

Here _we consider surfaces with all

eight

order one branch

points

distributed _on the unit circle.

Detailed

analysis

is made

possible by introducing

the

following mapping

defined in _a two- dimensional domain.

= 2

~A~

^{~~~ ~~~}

Since the

corresponding

Jacobian is

(B-A)/2,

the

mapping

is

regular

for A ~B and

A <B. We choose the former _case, for which the inverse

mapping

is of the form

A=-p+ fi

~~~

B=-p- p~+2-A

provided

that _p ^~ ₊ 2 A _m 0. In terms of

(3),

^{all branch}

points

^of the Weierstrass function

(2)

are

given by

bj,

~ ⁼ 4/

/

b~, ₄ =

bj,

~ ~~

b5

~ ⁼ 4/

/ _b~

~ ⁼

b5

_~.

Further, expressions (5)

and the condition

[b,[

=

I

(I

=

I to

8) give

the values of

A and B for which all

singularities

of

(2)

lie _on the unit circle. The solution is the domain

5~~= ((A,B) -2«A,B«2,A~B)

in the AB

plane,

I.e. the domain

5~~~ =

((p,A) -4p +2m0,

A

+4p +2«0,

A

-p2-2«0)

in the

HA plane,

both shown in

figure

I.

Among

^all

possible

minimal surfaces

corresponding

to the Weierstrass function

(2)

with branch

points only

_on the unit circle, we have _one two- pararneter, 4 one-parameter ^{and 5}

special

_«

Zero-parameter

» surfaces

(see

Tab.

I).

Since the

Weierstrass function R

(r )

is known for all these

surfaces, they

are

fully

described in _terms of the

Enneper-Weierstrass representation. Apart

from CLP

surfaces,

_no

single-parameter family

has been examined.

We shall consider in detail the

two-parameter family

of TPMES

corresponding

to the

Weierstrass function

~~~~~

~/r8- _{(A +B) r6+ (2}

+

AB) r4- (A +B) r2+1

^~~~

where A and B _are free parameters ^with

A~B and

-2<A,B<2

K is normalization

factor,

and

r is

complex

with

jr

_« I. The existence of such surfaces _was established in 1988

by

Koch and Fischer

[4b]. They

have orthorhombic symmetry ^and are

designated

OCLP because

they

_may be derived from the

tetragonal

CLP surfaces where the

bounding

unit is _a

right tetragonal prism

with

edges

_a, b and _c

[20] by elongation

_or

912 JOURNAL DE

PHYSIQUE

I N° 4

P~

t2

~

-la)

p

~

b)

i fi

~

A

~

f~

Fig. I. -rameter

for

all

possible ^inimal surfaces

(listed in

_Tab. I)

corresponding

eierstrass R IT ) =

~ withall

lyingon theunit

N/T ~ + 2 p T _~ + T ~ + 2 p T

Table I. The Weierstrass

functions

and the parameter ^domain

for

all

possible

minimal

surfaces

with

eight

order-one branch

points

_on the unit circle.

Minimal

surface(s)

Weierstrass function

R(r)

^{Parameter Position} in

domain

figure

~

A p ^~ 2-< 0 whole

~~~~

~~~~~

VT

~

+ 2 _MT^~ ₊ A

r~

_{+ 2}

p r~ + l

~~~~~~

r~+ pr~+1

^~

K

(r~+ I)~/r~+2(p _I)r~+1 ^~~"

^{~~ ~~}

(r2-1)~/r4+2(~

+

i)r2+1 ~~""

"° ~~

Scherk's first-order p

tower ^~

~ 4 ~

K

(r~

₊

_1)~ ^~~

Adjoint

_to Scherk's

P3

first-order tower

~~

r

(~2 ~)2 ~4

Se'f-adj°I"t

CLP

fi _PO

r + 1

914 jOURNAL DE PHYSIQUE I N° 4

compression

in either of two horizontal directions. As _a consequence, ^the

tetragonal family

of CLP surfaces is contained within the

family

of OCLP surfaces _{as a}

limiting

_case.

We consider the

properties

of _a finite minimal surface

piece

of the OCLP surface

(referred

to as the OCLP saddle

surface)

which _can be

thought

of _as inscribed in the

parallelepiped

with

edges

_a, b and _c. The OCLP saddle surface is bounded

by

four

straight-line segments (lying

_on

the

edges

of the

right prism)

and four curvelinear segments

(lying

in the bases of the

right prism).

The saddle surface is

produced by integration

of the Weierstrass function

(6)

_over the entire unit disk. We _note

that, by

_contrast, the FlhchenstUck

(surface element),

from which the infinite OCLP surface is

generated by

reflection _or rotation

operations

_over the surface

boundary,

is

produced by integration

_over the first

quadrant

^of the unit disk.

The Weierstrass function

(6) completely specifies

the first and second fundamental forms of the OCLP

family.

For

example,

the Gaussian curvature of _a

specific

surface _at the

point

corresponding

to w is

[5]

K=

)~

(l

+

(°'( ) (R(°')(

~.

In

principle,

the

Enneper-Weierstrass representation

of the OCLP

family (equations (I)

with function

(6)) readily

enables the

computation

of _a

specific

OCLP saddle surface

(corresponding

to

particular

values of the free parameters ^{A and}

B) prior

to

scaling (normalization

factor

K is

unknown).

In

practice,

these

computations

_are cumbersome since numerical

integration

is not _as

straightforward

_as it is often

thought.

In

particular,

numerical

integration

must be

preceded by

_a careful

analysis

of the behaviour of the

integrand.

An

appropriate

numerical method _must then be chosen and the

computational _output carefully

examined in order _to

establish confidence in the result. A distinction _must be made between _« well-behaved

» and

«

badly-behaved

_»

integrands [23].

When used

unadvisedly,

which is

encouraged by

the

ready availability

of

sophisticated

_{computer programs,} numerical

integration

_may lead _to serious

errors. Our

integrands

_{are «}

badly-behaved

_» since

they

_possess

eight singularities

_on the unit

circle

(I.e.

the

integrals

_are

improper),

which makes them

particularly

difficult _to handle.

Furthermore, large portions

of _our surfaces _are

generated by

small

regions surrounding

the

singularities,

and it is necessary ^to

employ

_some

procedure

for the elimination of

singularities,

or

finding asymptotic expansions

of the

integrals.

Numerical

integration

is

normally

used when

analytical techniques

fail. Before

using

numerical

integration,

it is

important

to _ensure that the

integral

in

question

cannot be

analytically

evaluated.

Parametrization of the OCLP

familyo

The

integrals

in the

Enneper-Weierstrass representation

of the OCLP

family (equations (I

with Weierstrass function

(6))

_are

hyperelliptic.

In

general, hyperelliptic integrals

cannot be

expressed

in terms of _a finite number of

elementary

_or

special

functions. In other

words, they

cannot be

analytically

evaluated. It is thus necessary to use direct numerical

integration

or

complicated

series

expansions.

Very occasionally,

_some

hyperelliptic integrals (known

_as

pseudo-hyperelliptic)

_can be reduced to

elliptic integrals,

I.e.

integrals involving

_square roots of

polynomials

of the third and fourth

degree

with distinct roots.

Further,

_any

elliptic integral

_can be

expressed

_as the _sum of

elementary

functions and of the three

special (non-elementary)

functions known _as

canonical forms of

incomplete elliptic integrals

of the

first,

the second and the third kind

[24- 25].

There _are several definitions of the normal form

(Legendre-Jacobi, Riemann, Weierstrass,

Carlson,

etc.

)

of which the

Legendre-Jacobi

form is the most common. The main

advantage

of these

special

functions is that their

properties

_are well-established and their

computation

is

easy. The evaluation of

elliptic integrals (I.e.

of

special

functions known _as

incomplete elliptic integrals

of the

rust,

second and third

kind),

is _now

mainly

based _on two iterative numerical

methods which are

highly specific

to the nature of the function

[25-29].

The traditional

arithmetic-geometric

mean method _was outlined earlier

[19],

and

employs

successive Landen and Gauss transformations of

elliptic integrals.

The method is

simple,

_converge

quadratically

and _can be

applied successfully

to

integrals

of the first and second

kind,

but is

unsatisfactory

for

integrals

of the third kind.

By

contrast, Carlson's method

[26]

is based _{on a new} definition of canonical forms and on successive

applications

of the

duplication

theorem. The method _can be

applied

to all three kinds of

integrals.

We will show that the

Enneper-Weierstrass representation

of the OCLP

family

of TPEMS involves

pseudo-hyperelliptic integrals. By reducing

them,

resulting elliptic integrals

_are

expressible

in terms of

incomplete elliptic integrals

of the first kind

(in

its

Legendre-Jacobi

normal form F

(@, k)) only.

F

(@, k)

is _a

special

function defined _as

~

~~~ ~~ ~

~/(l ^~~~l

_k~

_t~) i~ ~

where y = sin @. ^We assume that the variable

k,

known _as the

modulus,

is real and lies in the interval

lo, I]

and that the variable @, ^known as the

amplitude,

is _a

complex

number.

Reducing hyperelliptic integrals

is made

possible by

the

mapping (3).

In order to reduce

integrals (I)

^with

R(r)

ⁱⁿ form

(6),

_we consider three

complex integrals

~°~°'~~~°~°'°~~~ wo~/r8-

(A+B)r6+ (2+AB)r4- (A+B)r2+1

^~~~~

~~

'~ °~ two~/r~-(A+B)r~+(2+AB)r4- _(A+B)r~+1

~~~~ ~~~~~~

_{~ ~}

~o

N/r~ _(A

+

B) r~

₊

~~~B)

r4- (A

+

B) r~

+

~~~~

which

give

X*(")~1~0~~2

y* (w )

₌ I

(r~

+

r~) (8)

z*(w

₌ 2

ri

so that

(x,

_y,

z)

=

rite (x*),

Re

~y* ),

^Re

(z*)]

holds.

The

mapping

r'

(see

Tab.

II)

reduces

ro

and

r~

_so that

(8) gives

~ ~ ~

wi

_dt

~

~°'~

_~

~°'°~

~ ~~

2

~, ~/(t

+

2) (t

A

) (t

B

)

~

~*~~°

"

~*~"°~

_~

) ^w~

~/(t ₂₎ (/~

_A

_{) (t}

B

~~~~

with

w]

and w' defined in table II and

-2<A,

B<2 with A~B. The

mapping

r"

(see

Tab.

II)

reduces

rj

with the result

916 JOURNAL DE

PHYSIQUE

I N° 4

Table II. Reduction

of pseudo-hyperelliptic integrals

to

elliptic integrals.

Mapping Inverse Differential Fixed limit

mapping

~ ~'

~

~~~~2 $~~()+)j ^~~~~~ ~,~W~+I

r +2 r' -2 W( ~2

~

~~V~

_~~

+

~ _~~

(l

+~r"~

fi~~

+

~~

^{~ ~}

~ ~ 2

where

"

~2+B ^~ ~2+A ~~~/(2

+A)(2 +B)

with _a

~

p

and the limits

wl'

^{and w"} are

given

in table II.

We _see that the

integrals

in

(I)

which describes OCLP

family

can be reduced _to

elliptic integrals.

All that remains to be done is to express the results in terms of

Legendre-Jacobi

forms of the

incomplete elliptic integrals

of the first kind. We

integrate

from wo =

0,

and

by employing

the

procedure

described elsewhere

[18]

evaluate

integrals

in

(9) using integral

tables in

Byrd

and Friedman

[24].

The Cartesian coordinates _x, y and _z of _a

specific

OCLP saddle surface in the Cartesian coordinate system ^{shown in}

figure-2

are

given by

the

parametric equations

x(r,

_{w = Kg~ Re}

[F (arcsin P~ (w ), k~)]

2

y

(r,

_q~

)

= Kg~ ^Re

iF (arcsil~ P~ (w ),

_{k~ )1}

(10)

z(r,

_w

= zo Kg~ Re

[F (arctan P~(w ), k~)]

where

zo =

z(o)

= Kg~ F

arctan

,

/~

k~

Re stands for the real part ^{of the}

complex Legendre-Jacobi

function

F(@,k)

and

w is

complex

_w

= rexp

(I,

_w

),

with

(w

« I. For any

particular

values of the two free

parameters

^{A and B the}

multiplication

factors g~, g~ ^and g~, and the moduli k~, k~ and k~, as well _as zo are real constants, while

P~(w ), P~(w )

and

P~(w )

_are

complex

functions.

Thus,

all

components

^of

equations (lo) depend

_on the value of the _same two free real parameter ^and apart ^{from the} normalization factor _K, this

dependence

is described in

table II.

Since _we have

gz =

)gxgy kj=

I

-kjkj

and the

multiplication

factors and the moduli are related

(see

Tab.

IV),

of the six constants

z

y

b

a

Fig.

2. The Cartesian coordinate system anached to the OCLP saddle surface described by

equations

(10).

Table III.

Multiplication factors

_g~, moduli

k;

and

complex functions P, (w )

which appear in the

parametric equations for

the orthorhombic nvo-parameter OCLP

family of triply

periodic

embedded minimal

su~fiaces.

A and B _are

free

_parameters with

-2<A,

B < 2 and A

~

B,

while

w is

complex

^with _{w «} 1.

X y ^Z

2 2 2

~

fi fi~ N/(2+A)(2-B)

~ ~~~ ~-~ (2~~~)(/~B)

~

l 2w 2iw

1-w~

'~°'~ gxl+w~ gyl-w2 il+w2

918 JOURNAL DE

PHYSIQUE

I N° 4

Table IV.

Relationships

benveen

multiplication factors

_g; and moduli

k; appearing

in the

expressions for

the

parametric equations for

the orthorhombic nvo-parameter ^OCLP

family of triply periodic

embedded minimal

su~fiaces.

X y Z

/I-kjkj

~/I-kj(

^I

^I-kjk~

~'

l

k(

I

kj

^~

~/(l _{k()(I ()}

( g]

i

~

g(

i

~ g] g] g(

₊

g(

9y

^9x

(g~, g~,

g~, k~, k~ ^and k~) ^{which describe} a

specific

OCLP surface and appear ⁱⁿ

( lo) only

two are

independent.

There _are several _reasons

why equations (10)

_are

preferable

to the

Enneper-Weierstrass representation

in the

computation

of

specific

OCLP surface. It has been shown that the

integrals

in the

Enneper-Weierstrass representation

of OCLP

family

allow

analytical

evaluation

(resulting

in

parametrization ( lo)).

There is therefore _no need to evaluate them

numerically.

Equations (10)

_are

analytical expressions,

and the function

F(@,k)

has well-known

properties. Further, equations (10) require

the evaluation of

F(@, k)

which is much _more

straightforward

than numerical

integration. Finally,

in view of the fact that numerical estimation of

integrals

fails close to the

singularities, equations (10)

are

superior,

since the function

F(@, k)

is finite in the entire

complex plane [25],

_even at

singular points

of the Weierstrass function defined

by (5). Thus, equations (10) completely

obviate the need for

using

the

Enneper-Weierstrass representation,

I.e. avoid

integration

of the Weierstrass function in the

computation

of

specific

OCLP surfaces.

However,

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, computation using (10)

does not

by

^itself solve the

practical problem

of

computing

_a OCLP surface for _a

given edge-to-length

^ratio of the

bounding

unit.

Computation

of OCLP surfaceso

A OCLP saddle surface is bounded

by

a

right tetragonal prism

with

edges

_a, b and

c. For any

pair

^{of values of free}

parameters (

² _~

A,

B

< 2 with A

~

B),

the

equations ( lo) give

x(bi )

= a/2

y(bi )

=

b/2

z(bi )

= qj

(I la) x(b5)

=

a/2

y(b5)

₌ ^b/2

z(b5)

= q~

where

bi

^and

b5

are branch

points

defined in

(5),

and

[qi[

₊

_[q~[

= c.

Further,

it is not difficult to show that

Re

iF(arcsin P~(bi), k~)i

=

K(k~)

Re

[F (arcsin P~(b5), k~)]

₌

K(k~) ^(I16)

and

c = Kg~ ^Re

[F (arctan P~(I ), k~)]

= Kg~

K(k~) (llc)

where K stands for the

complete elliptic integral

of the first

kind, K(k

= F

(ar/2,

k

).

From

(10)

and

(11)

we obtain the normalization factor _K

~

g~

I(k~ )

_g~

~(k~

)

_g~

I(k~

^~

^~~'

^{~ ~~~~}

which is _a real and

positive

function of the free

parameters. Using (12)

and table

III,

K can be

easily analytically expressed

in terms of these parameters, so that _an

expression

for

K = K

(A,

B

)

is

readily

available.

Equations (12)

also define six

possible edge-to-length ratios,

of which

only

two are

independent.

Table III

gives readily

available

analytical expressions

for the

edge-to-length ratios, involving

the free parameters ^{A and} B.

They

are

independent

of the normalization factor _K, and

only depend (in

a continuous

manner)

_on the free parameters on

the one-to-one basis.

The constant zo in

(lo)

is _so chosen that the

parametric equations

describe _an OCLP saddle surface in the coordinate system ^{with the}

largest

absolute value of the Gaussian _curvature at the

origin (see Fig. 2).

It _was shown earlier

[20]

that in _case of CLP surfaces this

point

halves the

height

of

right tetragonal prism (I.e.

the

edge

_c is devided in the I _: I ratio and zo =

c/2). However,

for OCLP

surfaces,

the

edge

c is divided in the ratio

(see Fig. 3)

~

d~ K(k~)

~~~~

~~ F

(arcsin

,

~)

gx

where

d,

=

[q;[

and q; is

given

in

(lla). Analysis

of

(13)

reveals that D

= I for

A ₊ B

= 0

(I.e.

CLP

surfaces)

and D @ I for A + B W 0.

Parametric

equations (lo)

and

expressions

for the normalization factor

(12) give

x(r,

_q~

=

£ /

Re

iF (arcsil~ P~(w ),

_{k~ )1} 2

K(k~ )

y(r,

_w

)

=

] j

_Re

_{IF (arcsin Py(w ), ky)1 (14)}

( y)

z

(r,

_q~

= zo

/

_Re

_{iF (arctal~} P~(w ),

_k~

)1

thus

providing

the solution of _a

problem

which _can be formulated _as follows _{: «} find the coordinates of the OCLP saddle surface which is bounded

by

the

given right tetragonal prism

with

edges

_a, b and c". This _means that for any

specific

values of the

length

of

edges

a, b and _c there is

always exactly

one OCLP saddle surface

completely

described

by (12)

and

(14)

where k~, k~, k~,

P~(w ), P~(w )

and

P~(w )

all

depend

_on the values of the two free

parameters, ^{have the} same

meaning

_as in

(lo),

while _w

=

rexp(iw

with 0 _{« r «} I and 0 _{< w} « 2 _ar.

With B

= A

=

fi

_{where 0}

< A

< 2

(-

2

< A

< 2

),

the

two-parameter

^OCLP

family

with orthorombic symmetry is reduced to the one-parameter ^CLP

family

with

tetragonal

symmetry ^and

equations (10)

and

(14)

became

equations

of CLP surfaces in full agreement with results in

[20].

In

practice,

_we need _to find the values of the free parameters ^{A and B for}

given

920 JOURNAL DE

PHYSIQUE

I N" 4

a)

b)

Fig.

3. Four OCLP saddle surfaces

corresponding

to four different

bounding

units. The surfaces _are calculated

using equations (12)

and (14), and the parameters used in the

computation

_are

given

in table V.

Cl

di

Fig.

3

(continued).

922 jOURNAL DE

PHYSIQUE

I N° 4

a, b and _c, I.e. to solve _a system of two transcendental

equations

with two unknowns in

(12)

and

apply (14)

and calculate the coordinates for these calues of the free parameter. ^This

procedure

allows us to model the

given

structural data

belonging

to orthorombic symmetry ^and confined to the volume of _a

right tetragonal prism by

_a

specific

OCLP surface. Since

(12)

and

(14)

^contain no

adjustable

parameters, « yes-or-no »

modelling

is

possible.

In other

words,

we can

always

decide whether _{or not a}

given

^data set matches _a

specific

OCLP surface to a

prescribed degree

^of _accuracy.

We cannot recommend _a numerical method of

calculating

the free parameters ^{A and}

B

corresponding

t

given

values of _a, b and _c, because of the absence of reliable

general

methods for

solving systems

^of two and _more nonlinear

equations [29b].

In

particular, considering

the very

complicated expression

for the

edge-to-length

ratios defined

by (12),

it

seems that

employing

of multidimensional

Newton-Raphson

method

involving

derivatives is

not

promising

_even if

approximations

for the derivatives _are used. This

problem

deserves additional

analysis,

but in most

applications

it is sufficient to _use the iterative two-dimensional

fixed-point

(« ^successive substitution

»)

method

[30]

Uk "

d~l(Uk-l'~k-I)

_Vk

"

d~2(Uk-I, Vk-I)

where k

=

1, 2,

3, is the number of iteration steps ^and uo, vo are the initial guesses for the

roots. The

advantage

^of this method is its

simplicity

and

flexibility

in

choosing

of

fixed-point

equations

_u

= Ah

(u, v)

and _v

=

4l~(u, v) corresponding

to _a guven system ^of

equations fi (u,

_v

=

0

f2(u,

v

)

= 0.

The

disadvantage is, however,

that iteration does _not

always

_converge ; when it

does,

the convergence is slow.

The evaluation of functions F

(@, k)

is

computationally straightforward,

and

appropriate

programs are

easily

available either _as user-callable routines in

general

_purpose subroutine libraries _or built-in functions in various software

packages.

Carlson's and Bulirisch's

algorithms,

based _on Carlson's method and the

arithmetic-geometric

mean method, _are the most

satisfactory.

Almost all mathematical subroutine libraries

(such

as the NAG FORTRAN

routine

S21BBF(X,

Y,

Z, WAIL),

where WAIL is _a

machine-dependent constant)

_use

Carlson's

algorithm

which calculates

R~(x,

_y,

z),

the

incomplete elliptic integral

of the first kind in Carlson's canonical form. The _source code of its FORTRAN

implementation

_can

be,

for

instance,

found in reference

[29a].

^This

callable,

robust and

highly portable

routine rf

(x,

_y, z

)

enables the

computation

of F

(@, k)

^with

F

(@, k)

= sin rf

(cos~ @,

I _K^~

sin~

_{@, I}

).

Similarly,

the

computation

of

F(@, k) using

Bulirisch's

algorithm (the

_source code is available in reference

[29b]

as routine e12

(x, k~,

a,

b))

can be

performed

with

F

(@, k)

= e12

(tan @,

I _K~,

l, 1).

It should be noted that in both routines

changes

have _to be made

regarding

the

complex

aritlimetic. We _{use a}

simple

and very fast

algorithm

_on the

arithmetic-geometric

_mean

method,

because its _use for the evaluation of F

(@, k)

for

complex

values of the

amplitude

is well

documented. Its

only

known

unsatisfactory

behaviour is for k

~ l and

~

ar/2,

but these

cases do not arise here. The software for mathematical

applications (Mathematica, MathLab,

Maple)

contains

elliptic integrals

_as built-in functions. For

instance,

Mathematica has

elliptic integrals

and all related functions

(theta functions,

Jacobi and Weierstrass

elliptic functions), probably

based _on the

arithmetic-geometric

_mean method

(note

that

F(@,k)=

Elliptic

F

[@, k~]).

There is _no

published computation

of OCLP surfaces. In order _to

compute

^{four OCLP saddle} surfaces

(Fig. 3) corresponding

to different

lengths

_a, b and _c of the

bounding unit,

_{we use}

equations (12)

and

(14)

and

procedures

described above for

root-finding

in _a two-dimensional

non-linear system ^and

computing

the function F

(@, k).

The

length

of

edge

_a is assumed to be

unity,

and all

quantaties required

in the calculation _are

given

in table V.

Table V. Constants used in

computation of

the various OCLP saddle minimal

su~fiaces

shown in

figure

3.

Figure

A B _K

Edges

of

bounding

unit

a ₌

3 _a 1.34188481 1.34188481 0.55126521 b

= I

c ₌ I

a ₌

3 b 1.96999281 1.96999281 0.63302609 b

= I

c =

2

a =

3 _c 0.65299525 1.93971240 0.36523922 b

= 3/4

c ₌ I

a ₌

3 d 1.97469804 0.29508854 0.51759396 b

= 5/4

c ₌ 3/2

Acknowledglnent.

We _are

grateful

to Shell

Research, Amsterdam,

for

support.

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