VOL. 21, NO. 5
CHINESE JOURNAL OF PHYSICS
OVOBER 1989
The Second Area-Weighted Moment of Convex Polygons
K. Y. Lin ( ;fst; X $Q )
Department of Physics, National Tsing Hua University
Hsinchu, Taiwan 30043
(Received Sep. 14, 1989)
The generating function for the second area-weighted moment of
the number of convex polygons on the square lattice is derived. The
empirical formula of Enting and Guttmann is verified.
1. 1NTRODUCTlON
A self-avoiding walk on a periodic lattice has been used as a model of linear polymers1
A self-avoiding loop is a self-avoiding walk returning to the starting point. One of the
unsolved problems in Statistical Physics is to derive a generating function for the total
number of self-avoiding loops on a square lattice with a fixed length of perimeter.
A convex polygon on a square lattice is a special case of the self-avoiding loop such
that a straight line on the bonds of the dual lattice cuts the bonds of the polygon at most
twice. The generating function for the number of convex polygons was first derived by
Delest and Viennot in 1984.* Their derivation is extremely complicated. A much simpler
derivation was found by Lin and Chang in 1988.3 Another simple proof was given independently by Kim.4
Recently Enting and Guttmann5 proposed two empirical formulae of the generating
functions for the first and second area-weighted moments of convex polygons on the square
lattice :
p, 6) = 2x” fkpnk
n
= x4 (1 _ 12~2 + 50~4 - 76x6 +
42x8 - 48x” + 32x” )/(I - 4x’)“’
+ 4x8/(1 - 4x2 )S’2
(1)
P?(x) = Cx” f [k(k - 1)/21pnk
I?
= x6 [R(X)/(l - 4x2 )6 + S(x)/(l - 4x* )9’* I
399
(2)
400
where
= 2 + 5x2 - 2 2 4 x4 + 1306x6 - 3352x’ + 4536x” - 3424~‘~
R
S = - 2 9 x2 +
+ 1664~‘~ - 512~‘~
1 7 2 x4 - 356x6 +312x6 - 120x”
is the number of convex polygons with n steps and area k. The first generating
function (1) has been derived recently by Lin6 and the result agrees with their conjecture.
The purpose of this paper is to derive the second generating function (2). Hereafter we shall
refer to Lin’s paper as I. We shall follow the procedure and notations of I as close as
and
P,
k
possible.
11. THE FlRST SPEClAL CASE
Consider first the special case where the width at the top row equals the width of the
bounding rectangle. We define
(3)
G ”( x) = Fk [k(k - 1)/2lg, ix” = I: G;l,
m
where gn k is the number of polygons with steps n and area k, and Gi is the generating
function corresponding to all polygons whose width at the top row is m. We have
G;‘(x)
= x6(1 +3x2 +6x4 +10x6 + . . .) = x6/(1 :x2)3 .
(4)
The polygons whose top width is m+l consist of two types of polygons (see Fig. 2 of I). The
first type corresponds to adding one unit square to the right side of all polygons whose top
width is m, while the second type corresponds to adding one rectangle with area m+l on top
of other polygons. Therefore we have
G”m+l = x'(G; +G,J+ ??x~('~+~-'~)[G;; +G;(w~+l)+G~(m+l)~/2]
n= 1
(5)
which follows from the identities
(k+l)k/2 = k(k- 1)/2+k
(k + m + 1 )(k + ml/‘2 = k(k - 1)/2 + (m + 1 )k + (m + 1)42
.
When m = 1, we have
G; = x2(G','+G',)+x2(G;+2G;+G2)+x4(G','+2G;+G,)
L-.
(6)
K.Y.LIN
401
and therefore
G;’ = x6(1 +8x2
+ 5 x4 ) / ( 1 -x2)4 .
(7)
We rewrite (5) in the form
G;+l _ x2 GII, _ x G; x2(m+2-‘0 = R(~)
n
= @z+~)GL+~ -mx2Gh -(m+2)(m+1)G,+1/2+(m+l)mx2G,/2
.
(8)
Consequently we have
R(m+l)-x2&Z(m) = ( 1 -x’)G~+~ -~x~GII,+~ +x4Gi,
= (m + 2)Gk+2 - 2x2 (m + l)Gh+l + mx4 Gk
- (m + 3)(m + 2)G,+, /2 + (m + 2)(m + 1)x2 G,+l
-(m + l)mx4G,/2
= (mx2 + 2)Gk+, - 2x2 Gk+r + [m(m - 1)x2 /2 - 31 Gm+2
+2x2 G,+l .
(9)
The general solution of the recursion relation (9) is
G;
= [akm(m - l)(m - 2)(m - 3)/24 + b:m(m - I)(m - 2)/6
+clm(m - 1)/2 +dlm +el,J.zT +ct
(10)
where z, = x2 /( 1 + x) and ct denotes conjugate terms obtained from the others by the exchange of x with -x. Substituting (10) into (9), we get an identity for all m:
a’+m(m- l)[(l -x)(2m-1)-2(m-2)]/6+b:m[(l
-x)m-m+ll
+cl,[(l -x)(2m+l)-2ml - 2 x 4
= fjm3 +f,m” +flm+f
where
f3 = -xal/3 = -x5 /S(l +x>’
(11)
THE SECOND AREA-WEIGHTED MOMENT OF CONVEX POLYGONS
402
f2 = (1 +x)aI,/2-xb; = 7 x4 / 8 ( 1 +x)~
fi
=
-(3 + x&;/6 + bl, - 2xcl, = -x3 (11 - 3x + 2x2 )/S(l + x)’
f = (1 -x)c; - 2xdI, = x2 (2 - x + 3x2 )/4(1 + x)2
and the solution is
al, = 3x4 /8( 1 + x)’
bI, = -(ll - 3 x ) x3 / 1 6 ( 1 +x)’
2
2
cI, = (11 -6x+3x ) x /32(1 +x)’
d; = -x(5 + 9x + 1 5x2 + 3x3 )/64(1 +x)’
(12)
.
.
The coefficient e’+ is determined from
Gyz._ - G;’ = (e: + d:)(z+z_ - 2:) - [(CL + di)z: +
ctl
(13)
and we get
el, =
3x/32 .
(14)
Summing over m, we get
G” = =m Gil
= a:z4,/(1 - Z,)’ +bkZ:/(l -
+
=
d;z+/(l -z+)’ +
Z+)4
+ClZ:/(l -Z+13
++/tl -Z+> + Cf
2x6(1 -x2)2(1 -4x2 +7x4 - 6x6 + 3x8 - 2x” )/( 1 - 3x2 + x4 )’
= x6(2+ 18x2 + 1 1 2 x4 +582x6 +. . .>
(15)
K. Y. LIN
403
111. SECOND SPEClAL CASE
Consider next the special case where the top right-hand corner of the bounding
rectangle is also a corner of the polygon (see fig. 3 of I). We define
H” = ,ck [k(k - 1)/2]h, kx” = CH; ( x )
m
(16)
where
; IH;+n+l $- HA+,+Jm + 1)
HI+1 = x2 (HA +Hk ) + x2n=O
+ *mw (m+l)d21
+ G,(m + l)m/21
+n;,x 2(m+2-n)[G; + G;(m + 1)
(17)
.
We rewrite (17) in the form
S(m) =
H;+1 - x2H; - x2
3, *;+,+I
= (m + l)(Hh+l - Gk+l ) - mx2 (Hh - Gk ) + (1 - x2 )GL+I
- x2 G; - (H,,, - Gm+l )(m + 2)(m + 1)/2
+x2(*, - G, >(m + l)m/2
(18)
.
The recursion relation for Hi is
s(m + 1) - S(m) = H;+2 - H;+1 + x2 *;
= (1 - x2 )G”+2 - G;+l + x2 G; + Cm + 2)(Hm+2 - G;+2)
- (m + 1 )(I + x2 )(H;+1 - G;+l ) + mc2 W:, - G; )
-
wn+2 - %+2 )(m +
3)(m + 2)/2
+ (*,+I - G,+1 )(I +x2)(m +2)(m + 1)/2
- (H, - Gm )x2 (m + l)m/2
i_ ._
.
(19)
THE SECOND AREA-WEIGHTED MOMENT OF CONVEX POLYGONS
404
The solution of (19) is
% =
z; [A;m(m - l)(m - 2)(m - 3)/24 + B;m(m - l)(m - 2)/6
+ q?r(m - 1)/2 +Q?I +q +ct
+ w”’ [A’m(m - 1 )(m -2)(m - 3)/24
+ B’m(m - 1 )(m - 2)/6 + C’m(m - I)/2 + D’m + E’]
,
(20)
Substituting (20) into (19), we get an identity for arbitrary m:
zycf4m4 +f3m3 +f2m2 +fim +f)+ct+wm+l(g3m3 +g2m2 +g,m + g )
where
f4 = x3(1 +2x)A;/24(1 +x)’ = a:x3(1 +x-x2)/24(1 +x)
.
f3 = x3(1 + 2x)B;/6(1 +x)’ - (2 +5x + 2x2)x2A;/12(1 +x)~
= b;x3(1 + x - x2 ) / 6 ( 1 +x)-u;x2(2+3x-x2 +x3)/12(1 +x)
+ (A, -a+)x3(1 +x -x2)/2(1 +x)2
f2
=
x3(1 +2x)C+/2(1 +x)2 -x2(1 +2x)Iq/2(1 +x)2
+ x2 (12+23x + 10x2)A;/24(1 +x)~
= c[ex3 (1 +x - x2 )/2(1 +x) - blx2(1 +x -x2 +x3)/2(1 +x)
+a;x2(12+11x-x2
+ x3 ) / 2 4 ( 1 +x)+(B+ -b+)x3(1 +x-x2)/(1 +x)z
+(A+-a+)~‘(-3-4x+2x2
-2~~)/2(1+x)~ +x”(l+~-~2)/
4( 1 + 2x)( 1 + x)2
fl = x3(1 +2x)D1,/(1 +x)2 -x2(2-x)(1 +2x)Cl/2(1 +x)2
+x2(3+5x+4x2)B:,/6(1
+x)’ -A;x2(4+7x+4x2)/12(l
+x)2
= d;x3(1 +x-x2)/(1 +x)+c;x2(-2-x+3x2 -3x3)/2(1+x)
+b;x2(3+2x+2x2
- 2 x3 ) / 6 ( 1 +x)-a;x2(4+3x+x2 -x3)/12(1 +x)
(21)
K. Y. LIN
405
+(C+ -c+)x’(l +x-x*)/(1+x)* +(B+ -b+)2X2(-1 -x+x* -x’)/
(1 +x)2 +(A+ -a+)x2(-1 -x+7x2 -x3)/2(1 +x)*
-x7(2+x-3x2
f
+3x3 )/4(1 +x)*(1 +2x)
= E;x3 (1 + 2x)/( 1 + x)* - D;x* (1 - x>( 1 + 2x)/(1 + x)* + C;x4 /(l + x>*
= eix3 (1 + x - x2 )/(l + x) - dlX* (1 - 2X2 + 2X3 )/( 1 + X) + CiX4 (1 - X)/( 1 + X)
+(c+ -c+)x’(-I
-x+x* -x3)/(1 +x)2 +(B+ -b+)x2(-1 -x+3x2 -X3)/
( 1 +x)2 +(A+ -a+)2x4/(l +x)* - x7 (1 + x - 2x2 + x3 )/2(1 + 2x)(1 + x)2
A’(2w -1)/6 = -x*0/2
g3
=
g2
= B’( 2w -
1)/2 + A ’ ( 1 - w)/2 = -x*E +D(5w - 2 - 2x2 - x*/w)/2
-x8/(1 - 4x2)
g1
=
C’(2w - 1) + B’/2 + A’(w - 2)/6
= -x*F+E(4w-2-2x2)+D(8w-l-x2)/2+x6[5w-3(1+x*)
+x*/wj/(l -4x2)
g
= D’(2w - l)+C’w
= 2wD+E(4w-l-x2)+F(2w-l-x2)+x6(6w-2-2x2)/(1-4x2),
with
A+ - a + = x6/4(1 +x)(1 +2x)
B, - b, = x5 (-5 - 9x + 2x2 + 4x3 )/8( 1 + x)(1 + 2x)*
C+ -c+ = xs /8( 1 + 2x)’
From (22) we get
2
A: = 3x4(1 + x - x ) / 8 ( 1 +x)(1 + 2 x )
(22)
THESECONDAREA-WEIGHTEDMOMENTOFCONVEXPOLYGONS
406
q = -x3
(11 + 30x + 2x2 - 3 1x3 + 6x4 + 1 2x5 ): 16( 1 + x)(1 + 2~)~
c:, = x2 (11 + 49x + 50x2 - 41x3 - 33x4 + 100x’ + 60x6 - 48x’ - 32x8 )/
32(1 +x)(1 +2x)3
0; = x(1 +x)(-5 - 24x - 42x2 - 35x3 + 8x4 + 44x5 - 32x7)/64(1 + 2x1~
El = x(1 +x)(6 + 42x + 105x2 + 11 1x3 + 4x4 - 108x5 - 64x6 + 64x’ + 64x8 )/
64( 1 + 2~)~
A’ = -6x’O/(l - 4~‘)~
B’ = -9x’O /(l - 4x2 )2 - x8 (8 - 27x2 )/(l - 4x2 )5’2
C’ = x8 (-6 + 23x2 )/( 1 - 4x2 )5’2 +x6 (-2 + 1 3x2 - 25x4 + 1 Ox6 )/( 1 - 4x2 )3
D’ = x6 (2 - 9x2 + 6x4 - 2 x6 ) / 2 ( 1 - 4~~)~ +x8(-l +6x2 - 18x4)/
2( 1 - 4x2 )“2
(23)
Finally we calculate E’. We have
H’I = [(D; + E;)z+ + ct] + (D’ + E’)w
H;’ = [(C+ + 20; + E;)z: + ct] + (C’ + 20’ + E’)w2
(24)
It follows from (17) that
H;’ = x2 (HI’ +H;) +x2
n;,(H;+2 + 2H;+2 + H,,+2 )+x"(G','+2G;+G,)
Z = x2 i, (H;+, +HJI+l )
and therefore we have
H;’ -HI’ = x4 (GI’ + 2G; + G, ) +x2 zo(H;l+2 + H,,+2)
= x4(G;‘+2G;
+G,)+_?(H’+H-H; -H,)
= x6 (1 - 1 lx2 + 16x4)/2(1 - 4x2 )3,2 +.a?(3 - 3x2 + x4)/(l - x2 ~3
(25)
i. -
408
THLSECONDAREA-WEIGHTEDMOMENTOFCONVEXPOLYGONS
1V. GENERAL CASE
A convex polygon can be divided into two (top and bottom) polygons as shown in fig.
5 of I. The area K of the polygon is the sum of K[ (the area of the top polygon) and K~ (the
area-of the bottom polygon) and we have the identity
k(k - 1)/2 = k,(k, - 1)/2 + k,k, + k, (kb - I)/2
.
(30)
Following the same procedure of I, we have
= G” + 2zz2’GL h,,, + GI, hl, + G, h; ) +mf3(G;gm + GA p; + G,g; )
(31)
where
gi, =
m-2
2n
(m-n- 1)Gfi .
The calculation of P2 is very complicated. We shall leave the details in the Appendix.
The final result is
pz (x) = x6 (R/A” + S/A9’2 ) = x6 (2 + 24x2 + 21 8x4 + . . . )
(32)
where A = 1 - 4x2, R and S are defined by (2). Our result agrees with the conjecture of
Enting and Guttmann.’
ACKNOWLEDGMENT
I thank professor Guttmann for sending their preprint prior to publication and the
National Science Council for financial support.
K. Y.LIN
409
APPENDlX
The generating function P2 is derived in this appendix. It was shown in I that
gtrl
= -1 + [(I +xp +(l -xp l/2
g;, = -
0
; x(1 + xpm /4 - m(l - x2 )/8( 1
(Al)
+xy
-( 1 -x*)/8x(1 +xy
+ ct
042)
h?l
= -1x/2(1 + 2x)(1 +x)m-j +ct] -x4(1 +A1’2)A-1(w/x2)m
hl,
=
[
0
+Y1
;
+m+
a+
+r+l(l +xY fct
(A3)
a +mp
(w/x’ 1”
(A4)
where
.
(y = -x6@-’ + A-3’2)
0 = _-2x8,-2
Y = x4 ( 1 + 4x4 )/2A2 + x4 /2A3’*
&+ = x2(1 +x)2/4(1 +2x)
0, = -x(1 +x)‘(l
+ 3 x + 2 x2 - 4 x3 ) / 8 ( 1 +2x)’
y+ = x3(1 +x)‘(l
-2x)/4(1 +2x)*
.
We have
g; =
m-2
z
x-*/‘(m - n -
n=l
I)G; = 4:f4
+ bkf3 + c>f2 +
where
It follows from the identity
k;. Ykf, = m-2 (m -n - I)[(1
,,.$
+v)/(l +x)J”
d:fl + elf,, + ct
645)
THE SECOND AREA-WEIGHTED MOMENT OF CONVEX POLYGONS
410
=
m(l +y)/(x -y) +
[(l +v>2 - 2(1 +u)(l +x)
+ (1 +JJ)m(l +X)2-m l/(x -v)2
646)
that
fO = [-l-2x+mx+(i
+X)2-ml/X2
fk = m( 1 + x)/xk+l - (k + 1 )(l + x>2 /T8+2
+ (1 +
X)~-~ ii
0m
i=O i
(k + 1 - i)/xk+2-i
,
k>O.
(A7)
Substituting (A7) into (A5), we get
3x2/8(1 +x)“’ +
m
0
+2
x(1 +3x)/16(1 +x)”
3/32( 1 + x)“+~ + 3m(l - x2 )/64x(1 + x)“‘-l + cf.
648)
Similarly we have
,,”
=
m
m~-lx-hh*
n=1
649)
n
where
+ ctl + A’F4 n (w) + B’F, n(w) + C’F2 n(w) + D’F, n (w) + E’Fo n (w)
Fk,, (f) = p;[;j t”+P = ii(:) t”+k-i/(l - t)k+l-i
and the last equation follows from the identity
It can be shown that
1
In-
c
II=
1
x-~‘~F~,,,(~) =
k. tk-‘(1
- t)r-k-l f,, (t/g >
(A101
K. Y. LIN
411
where
fr,,
(t> .=
m-l
z
n=1
n
0 t” =
f(l
_t)-‘-1 -t”(l -t)-1 ;:
(m )
s=o r - s
r
[t/(1 - t)1’ ,
r> 0
(t - tm )/( 1 -
t)
fO,m
=
f 1P
= t/(1 - t)Z - tm+l /(l - tj2 - mtm /(l - t)
f 2P
= t2 /(l - t)3 - tm+2 /( 1 - t)3 - mtm+’ /(l - t>2 - (;)P /(l -t) .
After some algebra, we find
h; = [(l +xym IS4 m pi +
i=o 0i
ctl + (w/x’ )” i$o ;
0
4;
(Al 1)
where
.
P4
= uA: = -3x3(1 +x)/8(1 +2x)
P3
= u(B; +v,A;) = x2(1 +x)(5 +7x -6x2 - 12x )/16(1 +2x)2
P2
= u(C; +v,B; +v,A;)
3
= ex(l + ,y)( 1 + 4x + 7x2 + 4x4 + 32x5 + 32x6 )/32(1 + 2~)~
Pl
= u(D; + v, C+ +v2B: + v3A;)
= (1 +x)(3 + 21x + 57x2 + 79x3 + 12x4 - 76x5 - 32x6 + 32x7 )/64( 1 + 2x1~
PO
= U(E; + v,D’, + v2 Cl + v3B; + veA)+)
= (1 + x ) ( 3 + 2 1 x + 5 1 x 2 +45x3 - 54x4 - 178x* - 72x6 + 1 84x7 + 1 60x8)
/64x( 1 + 2~)~
U
=
VI
=
-x2/[(1 -2+)(x2 -z+)J = -(I +x)2/x(l +x-,x2)
z+/(l - z+) +z+/(x2 - z+)
= (1 +x -x2 +x3)/x(1 +x -x2)
THE SECOND AREA-WEIGHTED MOMENT OF CONVEX POLYGONS
412
“rl = z”, &(l - z+)-j(x2 - z+p
= [z+/(l -2+)(x2 -z+)l”[(l -z+)“+l -(x2 -z+)““l/(l -x2)
= [(l +x -xy+a -x3@‘1)]/(1 +x)(1 -x’)x”(l +x-x2y
q4
=
A’lw = -3x8(1 +A”‘)/A’
q3 = (B’ + A’v;)/w = -x6/A2 + ( - 1 + 6x4 )x~/A”~
q2
= (C’ +
B’v; + A’v;)/w
= x6(1 -2x2 - 2~~)/2A”~ +x6(1 - 10x2 + 14x4)/2A3
41 = (D’ +C’v;+B’v;+A’v;)/w
.
= 2x8(1 - 2x2)/A3 -x*(1 -2x2 +~x~)/A”~
40 = (E’ + D’v; + C’v; + Bb,’ + A’v:)/w
= -x4 (7 - 42x2 + 88x4 - 76x6 + 28~~)/2A”~
- x4 (7 - 42x2 + 94x4 - 104x6 + 64x8 )/2A4
v;
= [(I _ w)“+l _ (x2 _ w)“+l ]/(_x’)“(l -x.2)
1,; = -1 _A1’2/x2
11; = (2 - 7x2 + 2x4 + 3x2 A”’ )/2x4
1); = -(2 - 7x2 +x4 )/x4 - (1 - 3x2 + 2x4 )A1’2 /x6
1,; = [ 2 - 1 3x2 + 27x4 - 23x6 + 2-P + (5x2 - 1 5x4 + 5x6 )A”’ ]/2x8
2~2 = 1 _ 2x2 _ A”’
2w3 = 1 _ 3X2 _ (1 _x’)A’,‘z
2w4 = 1 -. 4x2 + zx4 _~ (1 _ 2X2 )A”2
(El + c)” + ct) + k“ + q,, + H” = 0
i--
K. Y. LIN
413
p. + ct = x4 (7 - 51x2 + 1 37x4 - 1 68x6 + 80x8 )/A”
p, + ct = (3 - 30x2 + 19x4 + 280x6 - 752x8 + 384xl”)/32A3
p2 +ct = x2
(1 +7x2 - 8x4 - 48x6 + 128x8)/16A3
p3 + ct = x2 (5 - 27x2 + 64x4 - 48x6 )/8A2
p,+ct
=
3x4/4A .
Substituting hh into (31), we obtain
2;G
mc2
nl
h"
m
(Al21
=;S
/I=0 'I
where (i = 0, . . . , 4)
si =
x2 x [x/(1 +xp ; p,+ct
m
0
.
s,, =
'i+lO
x2 2 [x2 /(l - x2
m
)I” 7
0
pi + cl
=x2 T2 [W/(1 +x)Jrn mi qi+ct
m
0
*
It follows from the identity
tm
= t’/(l -t)
k = O
= t2 (2 - t)/(l - t)2
k=l
= tk/(l - ty+*
k>
1
(Al3)
that
so = p,x6/(l
+x)*(1 +2x)+ct
= x6 (-6 + 87x2 - 21 5x4 + 358-P - 2456x’ + 6304x” - 601 6xL2 >
/16(1 -x2 )/A5
s, =
p,s6(2+4x+x2)/(1
(Al4)
+x)*(1 +2x)* +ct
_
THESECONDAREA-WEIGHTEDMOMENTOI'CONVEXPOLYGONS
414
= x6
(6 - 93x2 + 450x4 - 2767x6 + 1 1204x8 - 20208~‘~ + 15808~‘~
- 512x14)/32(1 --‘)A’
(,A1 5)
s2 = p2 x” (1 + x)2 /( 1 + 2x)3 + ct
= x8 (5 - 30x2 + 2 8 9 x4 - I 320x6 + 3296x’ - 4864x” + 4352~‘~
- 2048~‘~ )/l 6A6
(Al6)
s3 = p3xy1 +x)2/(1 +2x)4 +ct
= x’O(5 + 66x2 - 5 1 9x4 + 1 608x6 - 2592x8 + 2 1 76x1’ - 768~‘~ )/8A6
(Al 7)
s4 = p4xy1 +x)2/(1 +2x)5 +ct
.
= 3~‘~ (9 + 1 5x2 - 108x4 + 144x” - 64x* )/4A”
( ‘4 18)
ss = @o +&)x6/(1 -x2)(1 -2x2)
= x10(7 - 51x2 + 1 3 7 x4 - 168x6 +80x8)/(1 -x2)(1 - 2x2)A4
(A19)
s, = (PI + &)X6 (2 - 3x2 )/( 1 - x2 )(l - 2x2 )2
= x6 (2 - 3x2 )(3 - 30x2 + 19x4 + 280x6 - 752x8 + 384x” )
/32( 1 - x2 )( 1 - 2x2 )2 A3
6420)
s, = x6(1 -x2)(p2 +ct)/(l - 2x2j3
= x8(1 -x2)(1 +7x2 - 8x4 - 48x6 + 1 28x8 )/16( 1 - 2x2 )3 A3
(A21)
s, = @I3 + ct)x8 (1 - x2 )/(l - 2x2 )”
= ~‘~(1 -x2)(5 -27x2 +64x4 - 48x6 )/8( 1 - 2x2 )” A2
(A22)
s, = (p4 +ct)xlO(l -x2 )/(I - 2x2 )*
= 3.~‘~ (1 - x2 )/4( 1 - 2x2 )‘A
S,, = x8 (14 - 78x2 + 14.5.~~ - 1 02x6 + 38.x’ )/A” (1 - x2 )
(A231
K. Y. LIN
-x8(14
415
- 92x2 + 217x4 - 222x6 + 1 18x8 - 56x1’)/A9’*
(1 -x2)
(~24)
S,, = x8 (3 - 1 3x2 + 1 6x4 - 2x6)/A4 - xl0 (6 - 35x2 + 58x4 - 26x6 )
/(I - x2)A4 - x8 (3 - 19x2 + 36x4 - 22x ” )/A9”
+ xl0 (6 - 23x2 + 26x4 - 2x6 )/( 1 - x2 )A”’
s,, = xa (6 _ 29x* + 46x4 -
22x6)/04 -
x8
(A25)
(6 - 41x2 + 90x4 - 70x6 + 8x8)
/ASi2
(A26)
S,, = x6 (2 - 10x* + 1 8x4 - 6x6)/A4 - x6 (2 - 14x2 + 34x4 - 30x6 )‘/A9”
(~27)
S14 = 6x8 (4 - 19x2 + 26x4 - 12x6)/A4
- 6x8 (4 - 27x2 + 56x4 - 42x6 + 8x8 )/A”’
.
(A28)
where we used the following equations
= [2 - 5x2 - (2 - x2 )A”’ ] /2(1 - x2 )A1’2
-[l -2x2 - 2x4 - A*‘? ] /2x(1 - x2 )A*”
Z,‘” - l)[w/(l +x)1” = w’/(l +x - w)2
= [l -3x2 - (1 -x’)A”~
](I - 2x)/2x2A
mg*b(m - 1)/2J [w/(1 +x)Jm = w2(1 +x)/(1 +x - Iv)3
= [2 - 9x2 + 8x4 - (2 - 5x2 + 2x4 )A”’ ] /2x2 A312
+ [ - 1 +5x2 - 7x4 +4x6 +(l -3x2 +3x4yA1”1
c
m=2
/2x3 A3’*
in
0
3
[w/(1 +x)1”
= w3(1 +x)/(1 + x - w)4
= (1 _3x+4x3)[1 _5x2
/2x4 AZ
+5x4 -(1-3x2 +x4,A”‘l
416
THE SECOND AREA-WEIGHTED MOMENT OF CONVEX POLYGONS
00
c
m=2
m
0
4
[w/(1 +x11” = w4 (1 + x)/(1 + x - w)S
= [4 - 31x2 + 79x4 - 79x6 + 28x8
_ (4 - 23~~ + 41x4 - 27x6 + 4x* )A”’ I /2x4 A5’2
+ [-1 +4x2 + 8x4 - 44x6 + 42x* - 8x” + (1 - 2x2
-10x4 + 24x6 - 12x’ )A1’2 I /2x5 As’2
(~29)
to derive (A24) - (A28).
Substituting h,,, and hm ’ into (3 1 ), we get
00
2m
m
.
6430)
where
f4 =
a; ,
f3 = b; , f2 = cI, , f, = d: ,
f. = el, ,
2 ; Cl, h;, = 2E[cum(m-1)/2+flm+y][a+m(m-1)/2+
m
m=2
+ b + m +c+l [w/(1 +x)1” +2X [a+n~(m - 1)/2 +
),I
+ fi+rn + y, 1 [a+m(m - 1)/2 + b+m + c, 1 [x/( 1 + x)J 2’n
+ 2C [cr+m(m - 1)/2 + p+m + y+ I [a_m(m - 1)/2
m
+ b _ m +c
- j[x2/(1 -x’)j”’
+ct
(A31)
where
% = -x3 /4(x 5 1)
Using the identity
b, = x2 (3 k x)/8(1 f x)
c, = -(5x)/8 .
K. Y. LIN
[Am(m -
0
QA ;
1)/2 + Bm + C] [am(m - 1)/2 + bm + CJE
+3(uB+bA+2aA)
417
(~32)
+ [aA +aC+cA +2(aB+ bA+ b&l ‘:
0L
+ (bB+ b C + c B ) m + CC ,
we can express (A30) and (A31) in the form
(-433)
where (i = 0, . . , 4)
r.; 47 [x/Cl +xPm+ct
‘i+l5
=
‘i+*Cl
=si mx i
‘m=
i
m
.
0[x2/(1
0 [w/Cl
, m y
‘i+25 = t.2
-x2>lrn
+x)1" +ct
3
r. = -x*(1 +x)*(3+9x+8x2 - 4 x ) / 3 2 ( 1 +2x)*
rl
= x2 (1 + x)(7 + 21x + 23x2 + 23x3 - 2x4 )/64( 1 + 2x)*
r2
= -x3(1 +x)(19 +26x - 23x2 - 18x3 )/32(1 +2x)*
r3
= x4 (1 + x)( 1 - 2x)(23 - 9x- 92x2 + 24x3 + 24x4 )/16A*
r4
= -9x5 (1 - x - 2x2)/8A
4
so = x2(3 - 19x2 + 3 2 x + 16P)/16A2
SI
= -x2 (5 - 1 2x2 + 6x4 - 124x6 - 3x8)/32n(l -x2)*
-x2(1 -3x* - 9x4 + 47x6 - 4,u8 >/ 16( 1 - x2 )A*
32
=
x4(-39 + 101x* + 2 1 9 x4 + 3 9 x6 )/16(1 -x’)‘A
+x4(-l +24x2 - 91x4 + 100x’ )/S( 1 - x2 >A2
s3
=
3x4 ( 1 - 6 x ’ -x4 +30x6 +8x8 )/4(1 --*)A*
THE SECOND AREA-\YEIGHTED MOMENT OF COSVEX POLj’GOSS
418
- ~“(1
1+9x2-195x4 49x6) / 8(1-x’)*A
s4 = 3x6(-1-2x2+19x4) / ~(I-x*)~A
lo = -~~[5--12x*+Sx~i(5-12x~)A~~~]
I~ =
/ 16A’
x5[5+21x+1 lx*-29x3+Ix4+20xs+16x6)
+ (5+21x+11x2-29x3-60x4-12x5)A1’*]
/ 32(1-+x)‘A2
l2 = - x6[1 1--6x-21x2+6Ux3-28x4-32x5)
+ (1 1-6x-21x2+31x3-12x4)A”2] / 16(1+x j2A2
.
t3 = x7[(11-13x+9x2+169x3-221x4+6x5+120x6--Gx7)
+ (11--?3x+9x2i97x3-1Wx4+30xs)A1~2]
/ S(l-x2)‘A2
f4 = - 3x*[1--&+x2+12&-Qx4 I (1_6x+5~~i_?x~~.r~)A~‘~! ! :(!-x~)~A”*.
It follows from (A13) that
SlS = ro x4 / (l+x)‘(1+2X) + Cf
= - x6(3-10x2+d8x4+32x6) t 16A3
s;6 = rl x4(2+3x+x’) / (l+x)‘(l+2x)‘+
=
(A33)
cf
~~(1335,~*-1S3~~-33~~+1SQ0x*-336x~~
/ 33(1-x’)A4
(A39
s17 = r2 x4(1+x)? I (1+2x)3 + Cl
=
x8(107-690.~2+1635x4-2076x”+1616x8-576.r’o)
s,s = r3 x6(1+x)? / (1+2x)’ +
1
=
/ 16.I’
(A36)
Cf
x’0 (23+270x2-25S9x4+6576x6-7~SSx8k036xS-76Sx1@)
.S19 = r4 x8(1+x)* / (Ii-2~)~ + cf
I 8A6
(A37)
K. Y. LIN
= 9x’~(9i15x*-10sx4+l??x~~x~)
/ 4A6
419
(A3S)
sa = so x4 / (1-x2)(1-2x2>
=
x6(3-19x2+32x4+16x6) / 16(1-x2)(1-2x’)A’
(AW
52, = S) x4(2-3x2) I (1-x*)(1-2x*)2
=- x6(10-39x2+4Sx4-266x6+366x8+9x1@)
- x~(‘-9.~~-gx~+121x~-1:9x*+12x~q
I 32(1-x2)3(1-2x2j2A
/ 16(1-x’)2(1-2r2)2A2
(A40)
sp = s* x4(1-x2) / (1-2x2>3
=x8(-39+101x%219x4+39x6)
I 16(1-x2)(1-2x*)3A
+ xs(-1+24x2-91x4+100x6) / 8(1-2x2)3A2
(A40
su = s3 x6(1-x2) / (r-k2>4
= 3x10(1-6x2-x4+30x6+8x8)
-
SSJ =
=
x’“(11+9x2-195x4~9x6)
s4
/ 4(1-2~~)~A*
/ 8(1-~*)(1-2x*)~A
(~42)
x8(1-2) / (1-2x*)5
3x+-1-2x2+19x’) / 4(1-~3(1--2x*)~A
(A43)
Su =x6(5-13x2+4x4-8x6) I ~(I-x~)A”~ - x6(5-3x2-12x4) / S(l-x2)A2
(A441
Sz = x6(3-26x2+130x4-278x6+399x8+60x’?
(A45)
/ E(~-x~)~A~
- x6(3-14x2+63x4-28x6) / 8(1-~‘)A”~
Sn =x:(42-155x2-71x4+5x6) I 8(1-x2)A3
- x6(42-239x2+177x4+4-48x6+11~*) / 8(1-x2)4”’
Sn = -x5120-183x2+484x4-316x6+184x8
-.
L
(A46)
THE SECOND .4REA-N’EICHTED hlOSlfST OF CONVEX POLYGONS
420
- (30-113~~+2~s~4~6x~+120x*)~~‘~]
/ 2(1-xQ4
@47)
SB = 3x6[2-39x2+16Sx4-240x6+16Sx*-32x10
- (2-35x2+102x4-9Sr6+56xs)A’!2]
/ 2(1-x2)A4.
Substituting g, , g,,2, , g,” into (31), we get
+G;g;+G,g;)=C;'+G;-G"+
FSk
k=30
w!lere (i=O,...,4)
Sj+30 =
.
*imt3 ~~][X/(liX)lh + Cf
Si+35 = vimc3 ~)[x2~(1-X2)1m
u4 = (l+~)~u;/2 - 3x(l+x)aJ2 + 3x4/16 = 3xs/4
u3 =
(l+x)2f+2 - 3(1+x)(1+3x)aJ8 - 3x(l+x)b+A + xZ(l+3x)/32
=x3(5+3x)/8
112 = (1+x)%;/2 - (Ii-x)%JSx - (l+x)%J3 - x(l+x)c+E + 3x2(1+x)%4
= x2(3-x+x2) / 16
u, = (l+x)‘Q2 - (1+x)(1-x2)bJSx - (I-x?)cJS + 3x(1-x~)(l+x)/12S
= -x(3+4x+7x2+2x3) / 64
u. = (lsx)‘e;/2
- (l-x2)cJSx = (I+x)(l+2x+3xzj / 64
v4 = (l-x)%;/2 + 3x(1-x)0+/2 + 3x4/16 + CI = 3x6(l+xz) / 2(1-x2)”
~3 = x’(5+20x2+3x”)
/ 4(1-x2)l
v2 = x2(1+21*2+17x4+x6)
/ 8(1-x2)2
vi =x2(1+9x2+7x4-x6) / 16(1-~~)~
K. Y. LIN
421
Yg = -(1+5x2) / 32.
It follows from the identity
k=O
= t3(3-21) I (1-Q’
k=l
= r3(3-3t+f2) I (1-f)3
k=2
=
fk / (l-#+I
k>2
(A50
that
SK) = ug x6 / (1+x)*(1+2X)
+ ct
= X6(1+2JJcz+15X4+GX6) I 32(14)3A
(A511
S3l = 111 xyMr+X*) / (l+X)4(1+2X)z + Cf
= ~~(21-26~~-7O~~-251X~+35X~-4X~o) / 16(1-X’)‘A 2
(A52)
S32 = U2 X6(3+12X+15X2+6x3+X4) / (1+X)1(1+2r)3 + Cf
= x8(9+84.r2-633X4+-6S9X6+900X8+27X’o +4X12)
/ 8A3(1-X2)”
0453)
s33 = u3 x6(1+x)2 / (1+2X>? + Cf
= ~‘~ (27-5X’- 72~~+3Sx~)
Sg=
I lA4
W-I)
u4 2(1+x)2 I (1+2.# + Cf
3x”( 1+21x2-40x4+16x6)
/2As
6455)
= -x6(1+5x’) / 32(1-~~)~(1--2~~)
6456)
=
s 35 = v&x6 / (1-x2)‘( 1--zx*)
ss = Vl x6(3-5x2) I (l-x~)~(1--2X2)*
---
THE SECOND AREA-WEIGHTED MOMENT OF COti’VE); POLYGONS
422
= ~*(3+22r~-2~~~-3&.~+5~~)
/ 16(1-~~)4(1--2x~)~
(A57)
sj, = v2 x6(3-9x2+7x4) / (l-x2)2(1-2x2)3
= x~(3+53x~-131x4-3x~+11ox~i7x*4 / S(b-x2)4(1-2r~)~
(A5Q
S38 = v3 x6(1-x2) / (1-2x2)4
= x’“(5+20x2+3x4) / 4(1-x~)(1--2r2)4
(A59)
s,, = v4 xy1-32) / (1-2x2)5
=
3x’“(l+x?) / 2(1-x2)(1-2x2)5.
(AGO)
Finally we have
S; E Sg +
=
S24 + SJ~
3x”(l-3x2) / 2(1-~~)(1-2x~)~A
s; =s;
i-s*+su+sj,
= x’“(5-32~2+57x4-6x6+23x8)
/ $(L-~x~)~(~-x~).I~
= x8( -17+292r2-1791x4+1919x6-5321x8-365x’o~’~_718.l:‘~+712r’4-~,r’~)
/ S(1-~‘)4(1--2r~)~A~
S; = S; + S6 + S2, + S36
= x6( -?+3x2+37Ox4-2222x6+3~~x8+3O79x1o-S382x’2+S96x”-22~x~6)
/ 16A3(1-_x2j4(1-2x2)
= x6( -3-~9x2+~~23~4-9~S7x6+2244Ox8-9362~'o-212OSx'2-2272x'-+~S4x'6)
s; = S; + SJ1 + Sj2 + G;
=x6( 29-2OSx2+255x4+15~.~6-5316x8i33SSx1o+992x’2-192x’4)
--__
-~
._
/ 3’(1-_~~)~A~
/ 33(1-x’j4A4
L.’
423
I;. Y. LIY
S; f S, +
S26 +
SjO+
G;
= x6( 37-367x2t13SOx”-1S54x6-36Sxs-96x*?
- x6(3-11x2t63x4-232)
/ 16(1-x2)A”
/ S(1-s~~)~~
t ,~~(~-13X~+3~4k0x6-10x8-~?x1~ I A”’
p2=c; +c; + ?s,
k=o
=
.
s, + s2 + s, t
+ Sls +
s4
+ sn + s13 + s14
SIT + Sl* + Sly + Sjj + SX
= x 6(R /A6 + S /AgJ2)
Lvherz R and S are defined by (2).
--
6461)
r
424
THESECONDAREA-WEIGHTEDMOMENTOFCONVEXPOLYGONS
REFERENCES
1. P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ, N. Y.,
1979).
2. M. P. Delest and G. Viennot, Theo. Comp. Sci., 34, 169 (1984).
3. K. Y. Lin and S. J. Chang, J. Phys. A 21, 2635 (1988).
4. D. Kim, Dis. Math. 70, 47 (1988).
5. I. G. Enting and A. J. Guttmann, J. Phys. A 22,2639 (1989).
6. K. Y. Lin, Chin. J. Phys. 27, 235 (1989).