The T and CLP families of triply periodic minimal
surfaces. Part 2. The properties and computation of T
surfaces
Djurdje Cvijović, Jacek Klinowski
To cite this version:
Djurdje Cvijović, Jacek Klinowski. The T and CLP families of triply periodic minimal surfaces.
Part 2. The properties and computation of T surfaces. Journal de Physique I, EDP Sciences,
1992, 2 (12), pp.2191-2205. <10.1051/jp1:1992276>. <jpa-00246695>
HAL Id: jpa-00246695
https://hal.archives-ouvertes.fr/jpa-00246695
Submitted on 1 Jan 1992
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J.
Phys.
(1992)
2
France
I
2191-2205
1992,
DECEMBER
PAGE
2191
Classification
Physics
Abstracts
61.30
68.00
T and
2. The
CLP
02.40
The
Part
properties
Cvijov16
Djurdje
Department
(Received
May
7
The
can
fully
to
the
surfaces
the
it
first
with
to
any
tD
actual
We
surface
relationship
tD
real
time,
possible.
surfaces
related
ratio
axes
accepted
of
geometry
describd
all
parameters,
University
J992,
and
in
a
terms
surface
to
the
triply periodic
computation of
of
and
surfaces.
minimal
surfaces
T
Klinowski
Jacek
and
Chemistry,
of
Abstract.
be
families
Cambridge,
of
final form
in
minimal
tD
of
a
normalization
free
and
factor.
the
on
The
analytical
latter
ratio
We
parameter.
derive
crucial
is
Parametric
structures.
find
between
surface
the
the
free
corresponding
parameter
to
several
and the cla
prescribed
ratio
values
is
of
also
the
found.
cla
for
surfaces
of
make
cla
that
G-B-
it
the
axes,
three
and
to
are
z
list
exact
and
such
relationships
of specific
possible, for
compare
therefore
coordinate
approximation
rational
We
ratio,
surfaces
tD
show
their
matching
the
the
of
choice
a
for
tetragonal
of
cla
offer
expressions
equations for normalized tD
corresponding to any given value of
Straightforward physical applications
structural
data.
of this result and
discuss
the geometric
consequences
functions.
be approximated using elementary
A
can
to
IEW,
CB2
J992)
depends
surface
single
geometry,
4August
Cambridge
Road,
Lensfield
of
for
the
coordinates
of
ratio.
Introduction.
minimal
40 triply periodic
embedded
surfaces
(TPEMS)
derived
by various
mainly by group theory), have been
described
[1-3]. Over the last 20 years
TPEMS
have been applied in many
of the physical and biological
sciences [4]. They are
areas
useful
crystallographic
condensed
advocated
by
concept for the description of
matter,
a
as
Scriven [5], Mackay [6-7], Hyde and
[8], Mackay and
Andersson
[9] and Sadoc
Klinowski
[10]. In principle, the
Charvolin
and
translation
makes it possible to
symmetry of TPEMS
them to actual
match
I,e, to
surface
coordinates
with spatial
structures,
pattems of
compare
and to
establish
relationships
surface
between
properties,
and the
atoms,
such
curvature
as
volume-to-surface
ratio,
and
relationships
between
especially those
structural
properties,
provided by X-ray
diffraction
and
solid-state
NMR.
Modelling of
also
structures
suggests a
method
of quantifying
structural
changes by relating them to
transformations
(such as the
Bonnet
and
Goursat
transformations) of minimal
surfaces.
Finally, studies of interpenetrating
crystalline
with large unit cells and complicated
of cages
networks
and
channels
structures
exposed the
limitations
of
classical
crystallography,
showing the need for more
appropriate
Since
TPEMS
have
labyrinthine
they are likely to lead to simple
concepts.
structures,
descriptions of such systems. A rigorous
structural
classification
of all known
and hypothetical
TPEMS
will be of great
importance to materials
science
in general.
Approximately
methods
(now
2192
PHYSIQUE
DE
JOURNAL
N°
I
12
linking TPEMS and physical
often far
structures
are
surfaces
in experimental
application of minimal
described
mathematical
of them have been
empirically,
without
the precise
science is that
most
quantify all known
TPEMS,
establish
which
therefore
specification. It is
essential
to
to
geometric properties are related to the properties of physical systems, and to find a reliable
In
particular, it is imperative that approximate
computing their
coordinates.
method
of
computation of TPEMS to a prescribed degree of accuracy be systematically investigated.
A
TPEMS
is
described
by giving its parametric representation or the Enneper-Weierstrass
conditions
of a partial
differential
equation [I1representation, or by specifying the boundary
three-dimensional
completely
described
by the
is
l3]. In general,
surface
in
space
any
parametric representation of the form
Unfortunately,
rigorous. The
mathematical
from
main
arguments
obstacle
to
(X, y, Z)
coordinates
the
where
V), Y(U, V), Z(U, V)j
IX (U,
"
functions
are
wider
a
of
parameters,
two
and
u
However,
v.
in
most
cases
of
analytical expressions for the
TPEMS
Sometimes
coordinates
unknown.
parametric
are
a
expressed
in
representation of such
surfaces
of
functions
be
elementary
alone.
terms
cannot
For example,
parametric representation of surfaces of the T and CLP family involves special
functions
[14].
surface
Any minimal
is
described
by the following three complex integrals
x
=
Re
R(r)
iw
r~)
(I
dr
w~
i~
Re
y
=
iR(r)
(I
+
r~)
dr
(i)
w~
j~
Re
z
=
R(r)
2
dr
r
w~
known
thus
reduced
(r )
function
R
TPEMS
has
it
Enneper-Weierstrass representation.
these integrals. For this to
to solving
the
as
seems
considerable
known.
be
must
Since
a
possible to
practical
different
20
for
the
for
finding a
the complex
of
construction
surfaces
R(r)
function
the
construct
be
method
new
developed [15, 30],
been
problem
possible,
The
have
been
TPEMS,
any
minimal
(«
surface
of R (r ) for
a
described.
In
so
this
but
is
»)
specific
theory,
Weierstrass
may
involve
[16].
difficulties
by solving partial differential
equations of
equations,
but
this
method
is
not
easy and is
as
difficulties
increase
in the
rarely used. As rule, computational
order : parametric
represendifferential
equations.
Enneper-Weierstrass
tation,
representation and partial
minimal
Our primary
objective is the application of triply periodic
embedded
surfaces
to
interested
in developing
solid-state
chemistry. We are therefore
straightforward procedures to
surfaces
We will
demonstrate
such
with
actual
that parametric equations
structures.
compare
for
normalized
tD
surfaces
achieve
this objective.
Coordinates
the
second
The
tD
family
Following Koch
triply periodic
one
his
Schwarz
of any
order,
famous
[19].
of
surface
minimal
known
the
be
can
found
surface
minimal
TPEMS.
and
Fischer
embedded
[17]
we
minimal
problems conceming
The
tD
surfaces
use
are
a
symbol
the
surfaces.
free
The
first
refer
tD to
surface
tD
boundary [18],
variously
known
as
which
T
to
the
oldest
described
was
was
surfaces,
family
by Gergonne
known
subsequently
D~
surfaces,
solved
of
as
by
superman
N°
12
PROPERTIES
surfaces
as
a
«
and Gergonne
surfaces.
tetragonal
deformation
COMPUTATION
AND
of
Because
of their
the
surface
D
OF
tetragonal
2193
they
symmetry,
the
thus
;
»
SURFACES
T
abbreviation
thought
be
can
of
tD.
symmetries which could be generated in
straight-edged
polygons [17]. Among them is the
ways
skew
straight-edged 8-gon obtained by taking eight of the twelve edges of a tetragonal
parallelepiped (right tetragonal prism) with edges a, a and c. The solution of the Plateau
Problem
for
8-gon is a finite
such
minimal
surface
piece of the tD surface,
which
will
henceforth
be referred
surface (see Fig. I ). It is composed of eight
saddle
to as the tD
congruent
There
four
are
by using
different
six
by
related
parts
straight
line
four
only.
minimal
surface
A tD
neighbours by
I.
The
edges
twelve
tD
of
saddle
surface
individual
twofold
axes.
a
tetragonal
into
be
four
used
The
as
a
surface
corresponding
piece
of the
to
=
Since
there
for
different
We
offer
are
denumerably
different
surfaces
a
choice
eight flat points
ratios,
cla
of the
of
of
one
tD
tD
of
family
three
saddle
skew
many
tD
surfaces
straight-edged
correspond
such
surface.
to
parameters,
We
8-gons
shall
different
all
also,
of
two
related
are
their
to
spanning eight
surface
of
the
2
values
them
for
which,
surfaces.
one-parameter
are
way
a
tD
~
~
for
such
contains
by straight line
constructing an infinite
bounded
block
in
surface
saddle
parts
building
a
differ
tD
congruent
assembled
minimal
prism)
planes.
mirror
and
it
surfaces
(a finite
surface
right
can
saddle
:
different
skew
symmetry
divide
which
saddle
with
various
of
axes
segments,
segments
Fig.
surfaces
of tD
types
the
of the
real
first
prior
and
same
that
means
free
closely
time,
scaling,
to
This
parameter.
related
derive
can
the
to
the
analytical
2194
JOURNAL
expression
for
coordinates
relationships
sphere
their
the
on
which
~
surface
the
Gauss
the
standard
occurs
coefficient
a
parameters
in
with
The
either
(corresponding
14
A
(E
~x~
-
I, fl
-
0)
-
and
A
-Relationships
between
domains
of E, fl
Table
I.
The
parameter
and
p
2
l
p
2
ratio)
by
denoted
K,
real
K
is
K.
parametric
For
the
of
surfaces
value
same
factor,
so
free
of
w
=
u
+
tD surfaces.
respectively.
describe
the
),
2
~x~,
except
components,
w
«
I
derive
family
of
Kg~ Re
K,
are
(see Fig. 2a), All
parameter
the
~2
j~
~~~~
as
«
of
specific
a
surfaces.
analytical
an
We
expression
surfaces
saddle
tD
differ
is
same
and
»
values
tD
will
for
the
factor
by
which
matching
to
different
with
different
but
of
(I.e.
normalization
parameter,
crucial
of
the
of
factor
surface
show
to
that
it.
[14]
iF (q~x, kx)i
Kgx Re
=
=
with
the
free
known
coordinates
is
Kgx Re
=
z(u, v)
iv,
(-
+
of the
free
the
and
parameter,
representation
cases
fl4
comparison
a
tD
p8-1
the
unknown,
far
enables
y(u, v)
all
fully
-E
of
three
the
limiting
and
~
produced,
are
x(u, v)
where
fi)
all
to
of
A
corresponding to the same value
only by the multiplication constant,
and
structure,
function
of the
a
The
refer
often
fi
normalization
The
K.
a
Since
will
~~
multiplicity
a
given by (2).
~
I
function
=
which
parameters
2+A
differ
Weierstrass
We
~
+fi
surfaces
Two
cla
roots
parameter
1).
-
~/- A
~'~
2
determines
The
p
I
l,
~
image points
the
in the
(0, 1), (0, 1) and
are
E
IA
of
used.
fl
and
0, fl
E
~
surface.
saddle
appears
=
-
free
the
2)
~
0
E)
±
l13
(E
2
-
A
E
to
=
A
/$,
±
with
root
square
of them
be
can
under
I),
Tab.
(with
A
with
12
points
flat
of a specific
tD
stereographic projection
parameter
(see
the
E), (0,
±
E
parameter
The
of
N°
I
map
polynomial
the
related
are
0,
in
ratio.
cla
images
the
by
complex plane.
surfaces [15]
saddle
as
l
~
the
onto
tD
obtained
are
fl
of
/$,
(±
fl (0
the
to
unit
PHYSIQUE
DE
iF (q~x,
/fi)1
(4)
IF (q~~, k~)j
defined
in
components
tables
of
IIa,
IIb
and
w
equations (4) depend
is
on
complex
the
value
N°
PROPERTIES
12
OF
COMPUTATION
AND
SURFACES
T
2195
a) Multiplication
II.
constants
g~ and g~ and the moduli k~ and k~ used in parametric
equations for the tD su~fiaces expressed in terms of JFee parameters E, p and A (see text).
The amplitudes q~~ and q~~ in parametric
equations for the tD su~fiaces can be expressed in
Table
the
b)
various
and
g~
ways.
g~
defined
are
table
in
IIa.
I-E~
gx
fi
~~2
/~$~
fl~
I+fl~
2
/
l-E
~
~
~
~~~
~
~/-A
4
+
E~
~2
l+E~
~
~
p~
1+fl~
i~
2
~~+~~~
I_
fi
2
1~
~~/_A~
~~
a)
I
2
cos
w~
x
~
w
q~~
6(w/g~)~
+
l
2(w/g~)
+
I
+
2
z
(
w
sin~
cos~
q~~
~
~
w
w~
2(w/g~)~
+
l
2(w/g~)~
+
l
+
4(w/g~)~
~
w
2(w/g~)
+
(
(w ~/g~ )~
l
~/g~ )~
q~~
~
I
+
9g~ )~
w
b)
of
the
free
same
real
and
parameter,
relationship
the
between
them
~ ~~~ ~~
where
that
assume
we
the
variable
k~
k,
modulus,
the
of
elliptic integrals
form
~
~
~~~l
i~ ~/(I
either
and
E, p and A, is given in tables IIa, b. Re in (4) stand for real part of incomplete
of the first
kind F (q~, k),
defined
Legendre-Jacobi
normal
in its
traditional
as
t~)
is
real
and
lies
in the
interval
[0, 1]
and
the
amplitude, is a complex
number.
We will
often
various
properties of
use
F(q~, k) which
explained in detail in standard
[20-21].
texts
are
The multiplication
factors g~ and g~ and the moduli k~ and k~ in (4) depend only on the value
of free
(Tab.
IIa), and lie the
interval
(0,1).
Further,
simple trigonometric
parameter
transformations
allow
the amplitudes q~~ and
in
of equivalent
number
to
us
express
ways in
q~~
of complex
inverse
trigonometric
functions
(Tab. IIb). We shall use complex
terms
inverse
variable
sine
q~,
the
functions,
Quantitative
In
order
necessary
to
which
differ
for
of
characteristics
match
that
the
actual
different
the
structural
following
tD
of
the
free
parameter.
family.
describing
data
problem
values
be
solved
a
tetragonal
given
the
a
system
and
c
to
a
tD
dimensions
surface,
it is
of
right
a
2196
JOURNAL
tetragonal prism,
examine
find
closer
the
coordinates
of the
normalization
the
PHYSIQUE
DE
factor
appropriate
tD
and
establish
K,
N°
I
surface.
saddle
To do
that
relationship
the
we
between
12
must
the
properties of the surface.
Our
parametric equations (4) show (Fig. 2b) that, for any value of free
the
parameter,
point A on the real axis of the complex plane with
coordinates
(p, 0) (where p is defined in
surface.
This
Tab. IIa) is mapped into point A with
coordinates
(a/2, 0, c/2) of the tD saddle
derivation
of an
basis
of the
analytical
expression for the ratio of tetragonal
the
axes
which
(which we designate by x),
derived
from
Enneper-Weierstrass
cla
be
the
cannot
representation (Eqs. (I) and (3) alone. Thus, from (4) we obtain
ratio
cla
and
other
ll'~(fl
where
ll'~(fl )
and
are
2
V'z (P )
~
~
z
j
=
~~~
~
~
~~~
(w/g~)2
4
~
W
+
2
(w/g~)2
+
~
~
Table
IIa
shows
expressions (6) give ll'~(p
I and ll'~(fl
of the
complete elliptic integral of the
that
I
parameter.
terms
expressions (5)
ant
ratio
axis
a
=
c
=
where
g~, g~, k~ and k~
alone.
Using
or
A,
defined
are
IIa, the
table
so
K(k)
F
free
(w/2, k),
=
ratio
expressions
that
Kg~
K(k~)
2 Kg~
~~~
K(k~)
therefore
is
x
E, p
of the
value
any
kind,
first
are
tetragonal
the
for
=
=
In
in
(8)
table
be
can
for x
x
=
2
=
g~
K(k~)
gx
K(k~)
(8)
IIa, and depend on
easily expressed in
(E), x
x (p ) and
=
the
value
of
terms
x
=
x
of the
either
(A
are
free
of the
parameter
parameters
readily
available.
tetragonal axis ratio x is independent of the
normalization
factor, and
see
continuous
manner) on the free
only depends (in a
It follows
that x is uniquely
parameter.
characteristic
(an invariant) of a specific
surface
belonging to the tD family.
it to the tD surface
We verify (8) by ap
with A
14 which has the only known
value
2/2 ) [22] and limiting
of the cla ratio (x
2).
It
is
show
(A to
~x~ and A
easy
cases
therefore
We
that
the
fling
=
=
that
for
A
14
=
relationship
between
and
k~
l
~
~
I.e,
the
moduli
are
related
by
Landen's
K(k~)
K(k~)
l
+
k~ is
/~
/~
descending
transformation
/~
I +
2
which
gives [21]
N°
PROPERTIES
12
COMPUTATION
AND
OF
T
~' 1°~~
(a)
2197
SURFACES
z
y
Re (to)
x
fi
I
m
(w~
~~
A
C
B
~,
,
A
Re (to)
'
C
'
,
D
B
m
(c)
(to)
A
B
A
,
Re (to)
,
D
~
~
ifii
(a)
Construction
Fig. 2.
of a tD surface piece spanning
complex plane using parametric equations (I I). Shading
the
entire
unit
disc
including the boundary. (b) and (c)
surface by projecting
thick
lines
the unit
disc.
on
/(2
/),
an
8-gon by projecting a closed
that integration (I I) is
indicates
construction
of
characteristic
unit
disc in the
carried
parts
out
of
the
over
tD
l12. The limiting
required ratio (8) is x
of x
behaviour
be explained by using only the concept of tetragonal
distortion
can
as
follows.
Normalized
tD
surfaces
base
span the edges of right tetragonal prisms with the
same
edge a (which is conveniently used as a unit) but different edges c. In other words, the surfaces
Since
k~
1/2
=
g~/g~
(Tab. III)
and
=
the
=
2198
JOURNAL
Table
A
Limiting
III.
2) and for
-
PHYSIQUE
DE
for
of the various
constants
(corresponding to fl
0
values
E
I
-
and
-
gx
gz
N°
I
E
A
(corresponding
0
-
I
k~
E-1
0
k~
/
0
and
I
-
~x~).
-
0
E-o
p
to
12
x
K
~x~
2/w
0
~x~
0
2
tetragonally
are
length
of
increases
x
equation
xo.
numerically. We
x
express
x
it is not
we
[23]
However,
in
able
xo,
where
of
of
and
with
-
tD
surface
probably
E
the
of
a
given by (8) is
x
following procedure.
fl.
or
employ
applications,
most
limit
the
free
(A
parameter
increasing
c(A
there
root-finding
sufficient
To
-
)
~x~
-
2),
the
ratio
the
only
is
x
be
can
value
and
root
involve
the
of
only
solved
root-searching
real
one
of
method
and
the
which
approximate
to
prescribed
any
shorten
methods
numerical
best
it is
the
Since
for
transcendental,
is
the
either
to
necessary
is
x
a
0),
determine
to
adopted
approximation
rational
a
be
terms
known,
algorithm
=
have
in
(hence
infinity.
to
The
approaching
When
decreases
c
towards
important
is
It
distorted.
edge
the
interval,
interval
derivatives.
solving the equation
by a rational
function.
is
Brent's
x
xo.
=
We
use
form
4
p~ E'
(9a)
'
X
#
4
I
i
where
E is
the
free
parameter
determined.
By
with
0
E
~
need
to
have
found
rational
significant
figures.
employing
approximations for
Thus, for a given
numerical
solution
of the
be
~
l,
and
(see Tab.
xo,
coefficients
the
standard
the
x
~, ~'
0
IV)
with
determination
the
of
method
p~, q,
rational
numerical
a
of
free
(I
=
0, 1, 2, 3 and 4)
[23] we
interpolation
accuracy
is
parameter
better
than 7
reduced
to
a
quartic
4
£
,
with
given
coefficients
Table
IV.
intervals
Three
of the
0.05 £ E £
in
table
0,13
xo qi
E~
(9b)
o
=
IV.
approximations
rational
parameter
~Pi
o
of
the
form (9) for
ratio
x
(E) for
different
E.
£ L73
)
0.50 £ E £ 0.89
0.13 £ E £ 0.50
I
0
4.65nWWM2642
2
%15.10224088421
3
15023.24273163%
4
2.7l697576312VM
3.80339n64598255
1
1
4445.36501919f0&
95.016254824041
45.213715340127
1485%2176933356
213.073S84195259
1.051796760637337
5.37555S39100W09
10.97%3834344544
85.7994121752395
1.95526487007l338
N°
PROPERTIES
12
COMPUTATION
AND
OF
SURFACES
T
2199
8-gon
equation
surface
spanning
that A
attributed
14 has been wrongly
to the tD
I). By solving the above
by eight of the twelve edges of a cube (so that xo
numerically we find that the
5.3485782.
value is A
correct
The
expressions (7) and (8) give
We
note
an
=
formed
=
=
x
~
it is
so
that
clear
given a
equations,
following
and
normalization
the
if
and
c
that
then
K
expression
factor
related
is
K
for
normalization
the
K
factor
g~ and k~ are
and
where
Equation (10)
of
free
Until
III.
in
defined
in
and
only
for
tD
Ref.
[24])
in
for
dimensional
as
simplification gives
parametric equations (4) for
Further
the
tD
=
equations
Parametric
parametric
the
inverse
functions
sine
function
root
of
the
coordinates
for
the
the
complex
surface
correctly
coordinate
right-hand
of the
half
be
~~~~
=
normalized
disc.
In
obtained
0.8389223
a
normalization
known
(lo)
as
given
in
from
factor
a
=
amplitudes
the
useful,
w
=
u
it is
+
iv
q~~
coordinates
to
the
on
and
necessary
with
w
IIb). It is clearly
order
been
=
easily
which
root
square
the tD tetragonal
of the
represents
has
14
A
surfaces.
tD
and y (see Tab.
for both
root
unit
with
can
equations (4) with
practically
to be
complex variable
square
that branch
only
consider
x
for
( IO)
a
give
IIa
which
K
x
the
in
considered
both
and
table
a
parameter
now
0.8389223
For
is
length.
right tetragonal prism
of the
depend on the value of free parameter alone.
analytical expression for K in terms of the
an
specific value of the length of the edge a, K depends only on
changes continuously with it, with limiting values listed in
IIa,
table
expressions
Therefore, for
parameter.
value
table
(K
size
surface
K
K(kx)
=
~x
free
the
to
parametric
representation of
should be given units of inverse
surfaces
saddle
the
~
g~K(k~)
2
q~~
examine
to
w
I,
desirable
is I at I. In this
surface
for
which
choose
to
unit
entire
u
m
of complex
complex square
in equations
appears
terms
the
the
disc.
In
the
way,
0, I.e.
branch
same
what
follows
equation (4)
on
imaginary
correctly the surface on
sign of the expression for x.
represent
for the
axis
left-hand
the
of
we
and
part
the unit disc, it is
sufficient
to change the
Symmetry
require that for the
calculation
of the y
coordinate
surface be
considerations
the
correctly represented on the upper (I and II quadrant) or lower (III and IV quadrant) parts of the
Were it the only requirement, it could be
satisfied by a simple change of sign, just as
unit disc.
for
the
coordinate.
However,
expression
(4) for the y coordinate
gives a correct
done
the
was
x
representation in the I and the IV quadrant of the complex plane,
instead
of the I and
II quadrant.
To
problem,
solution
function
resolve
this
the
must
«rotate»
counterwe
in [14].
clockwise
by w/2. The required function for y is found using the procedure
described
The
equation for y, obtained from integral 231.00 in reference [20], which
satisfies
the
new
convention
conceming the branch of the square root is
of
y
where
the
signs
~y(W )
of
"
expression for
quadratic
all
~x
(I
W
).
is
q~~
terms.
(u,
v
derived
In
)
=
from
other
KgxRe iF (q~y, kx)i
that
for q~~ in the first row
the
relationship
words,
of
table
between
IIb
q~~
by changing
and
q~~
is
2200
JOURNAL
Further,
lengths
free
coordinates
right
to
edges
(conveniently
of the
c
parameter
( lo), the
a
convenient
it is
and
a
tetragonal
a
from
chosen
appropriate
of the
prism
coordinates
instead
polar
right tetragonal prism it
use
of
PHYSIQUE
DE
among
saddle
tD
I
N°
of the
is
Hence,
Cartesian.
find
to
necessary
A), from
and
spanning eight
given
for
value
of the
According (4) and
twelve edges of
(8).
E, fl
surface
the
12
of the
are
x(r,
~
0
±
=
(r,
y
0
±
=
z(r,
K(k~)
2
£
'
2
K(k~)
'
£
0
=
Re
2 K (k~
Re
[F (arcsin
ll'~(w ), k~)]
Re
IF (arcsin
~l~~ (<
w
I1)
), k~)
iF (arcsin ~l~~(w ), k~)j
where
4(w/g~)~
~/ w~
'~'~~~°
+
2(w/g~)~
+
l
~2
ll'~(w )
with
w
left
in
half
table
of the
sign
minus
0 +
cos
r
=
defined
to
F
(u
+
impossible
simplified
Consider
approaches
«
r
«
I
and
w
0
~
w
while
w,
k~, k~, g~
In the
there
iv, k)
is
no
be
cannot
the
separate
to
0
and
gz
equation for x, the plus sign relates to right half
complex plane. In the expression for y, the plus sign relates
arrive at parametric equations
lower half. In this way,
we
IIa.
family.
Unfortunately,
that
ir sin 0
=
for F (q~, k)
formula
addition
true
expressed in terms
real and imaginary parts,
of F
with
a
and
the
to
upper
of the
complex
and
minus
half
and
are
and
to
the
normalized
tD
amplitude,
so
(iv, k) only. It is
parametric equations (11)
(u, k) and
g~
sign
F
therefore
cannot
be
further.
limiting
the
behaviour
of
the
parametric
equations
when
the
free
parameter
appropriate
the
limit
0.
For
other
2) the
F (q~, k) exist,
but x
0,
because
(A
g~
g~
y
z
therefore
moduli k~ and k~ in equations (4) are 0 and I
respectively,
and
the
appropriate
functions.
Since the multiplication
functions F (q~, k) reduce to elementary
finite,
constants
are
functions
describe
surface
in
of
elementary
only,
and
with
the equations
expressed
terms
are
2 we
obtain
parametric
the cla ratio
approaching infinity. It is easy to show that for A
equations of Scherk's
surface z
In (cos y/cos x) [25], so that equations (4) derived
with the
limiting case of A
2.
be
extended
assumption A
2 can
to the
finite in the entire
complex plane, even at points defined by
The elliptic integrals in I I
are
points
(2). This
that
k)
is
finite
singular
of the
Weierstrass
function
(3). This is
F
(q~,
at
means
particularly important, in view of the fact that
numerical
estimation
of integrals fails
close to
singularities.
the
computations of tD surfaces are known. It is clear that
No expressions for the approximate
they can be deduced from equations (11) by an appropriate approximation of the incomplete
elliptic integrals of the first kind by elementary
functions.
This
procedure
deserves
additional
analysis, but an approximation for z immediately
itself.
Thus for A
2 the
values
suggests
of the
modulus
between
0 and I, but for A
8 the
of
value
k~ in the expression for z are
in terms
makes it possible to
the z
coordinate
of elementary
express
k~ is less than 0, I. This
functions
Since F(q~, k)
with a high
numerical
for small k, we
have
accuracy.
q~
its
limits
(see Tab. III).
=
=
Close
to
one
=
=
(A
limit
~x~
-
we
find
that
the
=
=
=
=
~
=
~
~
m
~
z
m
2
K(k~)
Re
arcsin
I
g/
N°
PROPERTIES
12
With
w
r
=
cos
ir sin
0 +
0
COMPUTATION
AND
part is
real
the
explicitly
)
z
m
OF
arcsin
SURFACES
T
found
2201
as
&
K(k~)
where
[~/(r~
&
=
For
A
entire
+
k~
cos~ 2
kj sin~
0 )~ +
disc,
unit
approximation
this
50
«
approximate
and
for A
000
«
coordinate,
the
gives
albeit
~/(r~
k~
numerical
accuracy
as
many as 15 significant
accurately, using only
to
less
cos~
2 0 )~
+
k) sin~
2 0
7 significant figures on
figures. It is even possible
function
the quadratic
to
the
to
w~
c
~
and
2 0 +
2 k~~
2
K(k~)
~
~
fi
hence
~
~K~k~)fi~~~~~~~
Computation
of tD
Since
[26] and photographs of models [27] of tD
surfaces
be regarded as
cannot
drawings [15] obtained using the Enneper-Weierstrass
representation are the only
of
their
computation.
Reference
Cartesian
for
[15] gives the
coordinates
3,
50
14 and
obtained by taking the real parts of complex integrals (our
sketches
quantitative,
known
case
2.01,
A
=
Eqs. (I)
and
(3))
surfaces.
defined
Computation of a
numerical
integration
on
the
unit
disc.
using the
Enneper-Weierstrass
representation
requires
evaluate
hyperelliptic integrals, and
be
carried
without
to
cannot
out
prior analysis of the
behaviour
of the
integrands and the choice of an appropriate
numerical
from
which are
method
available.
For direct
comparison, we have performed
among a number
such
described
computations for tD surfaces
computations are quite
below.
These
cumbersome
and require a careful
interpretation of data in the vicinity of singularities of the
Weierstrass
function,
since large portions of the
surfaces
generated by small regions surrounding these
are
singularities.
the
main
However,
weakness
of using the
Enneper-Weierstrass
representation
directly is that surfaces
corresponding to a specific value of A computed from them
be
cannot
applied, In other words, without the knowledge of K and x (neither of which can be deduced
from it), the
Enneper-Weierstrass
representation does not tell us how to generate a surface to
match
prescribed data. Also, it is impossible to
different
surfaces
tD
without
compare
By contrast,
normalization.
parametric
evaluation
of
functions
equations (11) require the
F (q~, k) and K(k)
which is much
easier, and yield all
information
required for matching, thus
making straightforward physical applications possible. We use them to compute tD saddle
for
surfaces
several
prescribed values of the axes ratio.
The
evaluation
of the
functions
F(q~, k) and K(k), is now usually based on the Gauss
algorithm of the arithmetic-geometric
and
Landen's
transformation
[21, 23, 28] and
mean
Carlson's
algorithm [29]. The latter is implemented in NAG,
SLATEC
IMSL
numerical
and
libraries.
However,
Gauss
the
algorithm
for the
because
its
evaluation
of
we
use
use
F(q~, k) for complex
of amplitude is well
values
In
documented.
order
demonstrate
the
to
simplicity of this algorithm,
outline
it
below.
arithmetic-geometric
The
of two
we
mean
numbers
and
complex
different
from
0
from
number
and
bo (where bo is any
ao
JOURNAL
DE
PHYSIQUE
i
T
tD
surface
2, N' 12,
DECEMBER
<992
79
2202
JOURNAL
ao),
by AG(ao,bo),
geometric (b;)
sequences
is
(a,)
denoted
and
~'
~<
defined
and
j
'
PHYSIQUE
DE
~~
the
as
'
~i
~
N°
I
limit
common
of
the
12
arithmetic
~i
~
1, 2, 3,
Both
convergent for any choice of ao
sequences
are
iteration
prescribed
difference
between
at the n-th
step (I,e,
accuracy
AG(ao, bo) can be approximated by a~
the required value),
b~.
K(k) for any particular value of k we start from
To
calculate
with I
=
and
bo. After
reaching
a~
and b~ is
less
the
than
m
After
iterations
n
/$.
bo
I
ao
=
=
have
we
ao,
~~~~
~j~
bo)
2 AG
F (q~,
calculate
To
k) for
particular
any
ao
by using
and
bo
I
=
descending
Landen's
of
values
k
from
start
we
/$
k(
=
and
q~
q~o
=
=
q~
sequence
k)
~
where
~
1, 2,
I
=
'
together
n,
~~
"~~~~
~
with
~~'
'
reached
at
iterations
n
the
arithmetic-geometric
we
have
AG(ao, bo)
2~
algorithm
15
and
is
figures
simple
We
and
l
+
j
k)
_,
After
prescribed
numerical
accuracy
Its only
algorithm.
mean
for k
is
coordinates
the
-
of tD
and
I
q~
saddle
2~ a~
quadratically)
converges
and
at the 4th,
reached
behaviour
have
fast (it
very
usually
is
unsatisfactory
computed
known
~~
~~' ~~
~
The
~~~~
...,
is
accuracy
to 7
'~
~
8th
-
w/2,
but
so
that
step, respectively.
iteration
these
do
cases
for x
1, 0.5
in the
calculation
surfaces
not
and
0.25.
are
given
arise
The
=
of
edge
Table
a
V.
different
is
assumed
be I
to
and
all
Constants
used
in the
of
axis
ratio
values
the
quantities
required
computation of
the
various
saddle
tD
x.
=1
=o.5
k
5.3485782
43.91463368
569.960o6523
K
0.71579938
1.054279364
1.899815152
0.85894265
0.543284378
0.289183541
0.44045702
0.15094129
0.041886923
0.36208944
0.593650606
0.676894704
0.19400238
0.022783273
0.00175451429
minimal
in
here.
length
table
V.
su~fiaces for
N°
12
PROPERTIES
VIa-c
pieces
unit
of the
are
disc
with 0
coordinates
the
coordinates
only the
asymmetric units,
list
Tables
surface
r
«
ro
coordinates
1/8
and
=
alone.
can
a
Table
and
n~
of
a
Ho
of
obtained
by
the
rows
I
in
b
w
eighth
one
each
for
mm.
«
the
The
=
n~ ro
w/16
(n~
tD
saddle
(no Ho)
cos
1,
=
8
and
Z
x
of
each
(r,
b
2203
SURFACES
T
sector
is
divided
in~
ro sin
(no
These
surface.
saddle
) computational
using
domain
8
a
x
5
minimal
is
a
sector
grid
where
b)
I,
=
3
no
x
=
Ho )
4).
0, 1,
It
is
clear
that
be
can
the
obtained
tables
different
three
c)
0.5,
=
4
x
value
same
in
by
tD
symmetry
of x
by
VIa-c
su~fiaces.
considerations
different
but
values
of
differ
in
a.
columns
The
0.25.
=
5
6
7
8
o
1
2
3
4
a)
1
2
3
the
...,
surface
coordi~ates of
no, a)
+
...,
surfaces
corresponding to
multiplying the
coordinates
Cartesian
VI.
=
of
and
OF
given by
are
(n~, no )
complete
Coordinates
b£
w
grid points
of
w
with
and 0
I
COMPUTATION
AND
4
S
o
i
z
3
4
b)
6
7
8
2204
JOURNAL
I
N°
12
(continued).
VI
Table
PHYSIQUE
DE
2
1
5
4
3
6
8
7
o
i
z
3
4
C)
Acknowledgements.
We
are
Carlson
grateful
to
of
State
Iowa
Dr.
C. Briggs of
University for
the
British
Council
for
conceming
discussions
support,
the
and
properties
to
and
B.
C.
computation
Professor
of
elliptic integrals.
References
[I
SCHWARz
A.,
H.
Gesammelte
Abhandlungen (Verlag
Mathematische
Julius
Springer,
Berlin,
1890)
vol.1.
[2]
A. H.,
SCHOEN
Report
[3]
[4]
[5]
[6]
[7]
[8]
[91
Infinite
No.
Periodic
Minimal
Surfaces
Without
Self-intersections,
NASA
Technical
(1970).
Phys. Colloq.
D-05541
TN
and KOCH E., J.
France
51 (1990)
C7-131-147.
S., HYDE S. T., LARSSON K, and LIDIN S., Chem. Rev. 88 (1988)
221-242.
SCRIVEN L. E.,
Nature
266 (1976) 123.
MACKAY A. L.,
Nature
314 (1985)
604-606.
MACKAY A. L., Physica 131B (1985)
300-305.
HYDE S. T. and
ANDERSSON S., z.
Kristallogr. 168 (1984) 221-254
225-239.
170 (1985)
MACKAY A. L. and KLINOWSKI J., Computers and
Mathematics
with
Applications 12B (1986)
FISCHER
W.
ANDERSSON
803-
824.
[10]
[ll]
[12]
SADOC
DO
and
G.,
CARMO
NJ,
[13]
[14]
[lsl
[16]
[17]
[18]
[19]
J. F.
DARBOUX
M.
J., J.
CHARVOLIN
Thdorie
P.,
Gdndrale
des
Phys.
Geometry
Differential
France
Surfaces,
of
(1986) 683, 48 (1987) 1559.
(Gautier-Villars,
Paris, 1887).
and
Surfaces
(Prentice-Hall,
Engelwood
47
vol. I
Curves
Cliffs,
1976).
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