- Email: [email protected]
On a physical characterization of the surface of an ellipsoid
Inr.1. EngngSn’. Vol. 28, No. 11, pp. 1205-1208, Printedin Great Britain.All rightsreserved
OcL?&7225/w
1990
$3.00 + 0.00
Copyright@ 1990PergamonPressplc
LETTERS IN APPLIED
AND ENGINEERING
ON A PHYSICAL CHARACTERIZATION SURFACE OF AN ELLIPSOID GEORGE
SCIENCES
OF THE
DASSIOS
Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, and Institute of Chemical Engineering and High Temperature Chemical Processes, GR 261 10 Patras, Greece Ab&aet-A simple expression is derived which furnishes the surface area of an ellipsoid in terms of the inertia density dyadic and the electric polarizability dyadic of the reciprocal ellipsoid.
Consider the triaxial ellipsoid
where al > a2 > a3 > 0. Depending on the parametrization of the ellipsoid (1) there are many known expressions for its surface S [l-4]. Their common feature is that they are all expressed in terms of elliptic integrals. One of these expressions, given by Bowman (formula (40) in [l]) is the following
(2) where
and e,, e2 are the two eccentricities e
The
1
Je=z
e 9
2
JhFz
al
a2
substitution t = er(x + 1)-l”
transforms the two elliptic integrals in (2) as follows
and
(4) (5)
1206
G. DASSIOS
In order to characterize dyadic
the integrals in the right hand side of (6) and (7) we consider the f& 8 %3 %*8 $ a=- %I@ %I++-, 4 4 a3
(8)
associated with the quadratic form (1) and its reciprocal dyadic [5] Q=a~*~~f,+a~~~~*+a~~~~~3,
(9)
associated with the reciprocal quadratic form afxf + a$$ + a’$: = 1
In fact, the form (1) is represented
while (10) is represented
(10)
by r*O*r=l,
(11)
r*Q*r=l.
(12)
by
The nth semiaxis of the reciprocal ellipsoid (12) is equal to the inverse of the nth semiaxis of the original ellipsoid (11). Furthermore, the reciprocation leaves the main eccentricities invariant. We consider the following elliptic integrals for the reciprocal ellipsoid (12) (13)
(14)
Then the area formula (2), in view of (6) and (7), assumes the form S=2na$+2na’a2j:,-2n
(a$ - aZ)(af - a3 p 1.
a3 The
ala24
four elliptic integrals Z& ZY,n = 1, 2, 3 are connected
(15)
by the algebraic relations [6]
1: + fi + I”: = a,a2a3
(16) (17)
Upon invoking Relations (16), (17) formula (15) yields the symmetric expression S=&
[(a; + a:)i:
+ (a: + a:)fi + (a: + az)fi].
(18)
The normal form of the inertia dyadic of the ellipsoid (11) is given by [7]
M = $[(a$ + az)f& 63 iI1 + (a: + a:)$
@ ii2 + (a: + a$)f3 631k3],
(19)
where V = +f a1a2a3
represents the volume.
(20)
Letters in Applied and Engineering Sciences
1207
On the other hand, the normal form of the polarization dyadic of the reciprocal ellipsoid (12) is given [8] in terms of the elliptic integrals (13), (14) by
(21) where p=_- 4Jr 1 3 v2a3
(22)
stands for the volume of the reciprocal ellipsoid. If we denote by P,,, n = 1, 2, 3 the principal polarizations 4R 1 fi=-----_ 3 v+&
of (12) then (21) yields
n = 1, 2, 3.
(23)
Similarly, if M,, n = 1, 2, 3 stands for the principal moment of inertia of (11) then formula (18) implies (24) The electric polarizability
tensor for the reciprocal ellipsoid is defined by d=QI+fi
(25)
I being the identity dyadic. The inverse of (25) furnishes (j-l=
fl@ v
Therefore,
92 60 $ ~
fl +
PI
+
P
+
%3 69 %3 -
4
+
v
+
4
(26)
*
formula (24) is written as S=
--&M:i2-’
where the indicated operation between the two dyadics is the double contraction Finally, if we introduce the inertia density dyadic
(27) [5].
M,,=; for the ellipsoid (ll),
and the electric polarizability
(28) density dyadic (29)
for the reciprocal ellipsoid (12) we obtain the formula S = 10nMd : ij,‘.
(30)
Expression (30) provides a complete physical characterization of the surface area of any ellipsoid as the double contraction of its inertia density with the electric polarizability density of the reciprocal ellipsoid. It is obvious that (30) can also be used to relate the polarizability of an ellipsoid in terms of the inertia and the surface area of the reciprocal ellipsoid.
REFERENCES [l] F. BOWMAN, Introduction to Elliptic Functions. Dover, New York (l%l). [2] B. C. CARISON, Some inequalities for hypergeometric functions. Proc. Am. Mafh. Sot. 17, 32-39
G. DASSIOS
1208
[31 M. S. KJAMKIN, Elementary approximations to the area of N-dimensional ellipsoids. Am. Math. Monthly 78 280-283 (1971). D. H. LEHMER, Approximations to the area of an n-dimensional ellipsoid. Can. J. Math. 2, 267-282 (1950). C. E. WEATHERBURN, Advanced Vector Analysis. Open Court Publishing Company, U.S.A. (1948). G. DASSIOS, Second Order Low-Frequency Scattering by the Sop Ellipsoid. SIAM J. Appl. Math. 38, 373-381 (1980).
[71 L. D. LANDAU and E. M. LIFSHITZ, Mechanics Pergamon Press, Oxford (1960). 181G. DASSIOS and L. E. PAYNE, Estimates for low-frequency elastic scattering by a rigid body. J. Elasticity 20, 161-180 (1988). (Received 13 December 1989; accepted 17April 1990)
OcL?&7225/w
1990
$3.00 + 0.00
Copyright@ 1990PergamonPressplc
LETTERS IN APPLIED
AND ENGINEERING
ON A PHYSICAL CHARACTERIZATION SURFACE OF AN ELLIPSOID GEORGE
SCIENCES
OF THE
DASSIOS
Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, and Institute of Chemical Engineering and High Temperature Chemical Processes, GR 261 10 Patras, Greece Ab&aet-A simple expression is derived which furnishes the surface area of an ellipsoid in terms of the inertia density dyadic and the electric polarizability dyadic of the reciprocal ellipsoid.
Consider the triaxial ellipsoid
where al > a2 > a3 > 0. Depending on the parametrization of the ellipsoid (1) there are many known expressions for its surface S [l-4]. Their common feature is that they are all expressed in terms of elliptic integrals. One of these expressions, given by Bowman (formula (40) in [l]) is the following
(2) where
and e,, e2 are the two eccentricities e
The
1
Je=z
e 9
2
JhFz
al
a2
substitution t = er(x + 1)-l”
transforms the two elliptic integrals in (2) as follows
and
(4) (5)
1206
G. DASSIOS
In order to characterize dyadic
the integrals in the right hand side of (6) and (7) we consider the f& 8 %3 %*8 $ a=- %I@ %I++-, 4 4 a3
(8)
associated with the quadratic form (1) and its reciprocal dyadic [5] Q=a~*~~f,+a~~~~*+a~~~~~3,
(9)
associated with the reciprocal quadratic form afxf + a$$ + a’$: = 1
In fact, the form (1) is represented
while (10) is represented
(10)
by r*O*r=l,
(11)
r*Q*r=l.
(12)
by
The nth semiaxis of the reciprocal ellipsoid (12) is equal to the inverse of the nth semiaxis of the original ellipsoid (11). Furthermore, the reciprocation leaves the main eccentricities invariant. We consider the following elliptic integrals for the reciprocal ellipsoid (12) (13)
(14)
Then the area formula (2), in view of (6) and (7), assumes the form S=2na$+2na’a2j:,-2n
(a$ - aZ)(af - a3 p 1.
a3 The
ala24
four elliptic integrals Z& ZY,n = 1, 2, 3 are connected
(15)
by the algebraic relations [6]
1: + fi + I”: = a,a2a3
(16) (17)
Upon invoking Relations (16), (17) formula (15) yields the symmetric expression S=&
[(a; + a:)i:
+ (a: + a:)fi + (a: + az)fi].
(18)
The normal form of the inertia dyadic of the ellipsoid (11) is given by [7]
M = $[(a$ + az)f& 63 iI1 + (a: + a:)$
@ ii2 + (a: + a$)f3 631k3],
(19)
where V = +f a1a2a3
represents the volume.
(20)
Letters in Applied and Engineering Sciences
1207
On the other hand, the normal form of the polarization dyadic of the reciprocal ellipsoid (12) is given [8] in terms of the elliptic integrals (13), (14) by
(21) where p=_- 4Jr 1 3 v2a3
(22)
stands for the volume of the reciprocal ellipsoid. If we denote by P,,, n = 1, 2, 3 the principal polarizations 4R 1 fi=-----_ 3 v+&
of (12) then (21) yields
n = 1, 2, 3.
(23)
Similarly, if M,, n = 1, 2, 3 stands for the principal moment of inertia of (11) then formula (18) implies (24) The electric polarizability
tensor for the reciprocal ellipsoid is defined by d=QI+fi
(25)
I being the identity dyadic. The inverse of (25) furnishes (j-l=
fl@ v
Therefore,
92 60 $ ~
fl +
PI
+
P
+
%3 69 %3 -
4
+
v
+
4
(26)
*
formula (24) is written as S=
--&M:i2-’
where the indicated operation between the two dyadics is the double contraction Finally, if we introduce the inertia density dyadic
(27) [5].
M,,=; for the ellipsoid (ll),
and the electric polarizability
(28) density dyadic (29)
for the reciprocal ellipsoid (12) we obtain the formula S = 10nMd : ij,‘.
(30)
Expression (30) provides a complete physical characterization of the surface area of any ellipsoid as the double contraction of its inertia density with the electric polarizability density of the reciprocal ellipsoid. It is obvious that (30) can also be used to relate the polarizability of an ellipsoid in terms of the inertia and the surface area of the reciprocal ellipsoid.
REFERENCES [l] F. BOWMAN, Introduction to Elliptic Functions. Dover, New York (l%l). [2] B. C. CARISON, Some inequalities for hypergeometric functions. Proc. Am. Mafh. Sot. 17, 32-39
G. DASSIOS
1208
[31 M. S. KJAMKIN, Elementary approximations to the area of N-dimensional ellipsoids. Am. Math. Monthly 78 280-283 (1971). D. H. LEHMER, Approximations to the area of an n-dimensional ellipsoid. Can. J. Math. 2, 267-282 (1950). C. E. WEATHERBURN, Advanced Vector Analysis. Open Court Publishing Company, U.S.A. (1948). G. DASSIOS, Second Order Low-Frequency Scattering by the Sop Ellipsoid. SIAM J. Appl. Math. 38, 373-381 (1980).
[71 L. D. LANDAU and E. M. LIFSHITZ, Mechanics Pergamon Press, Oxford (1960). 181G. DASSIOS and L. E. PAYNE, Estimates for low-frequency elastic scattering by a rigid body. J. Elasticity 20, 161-180 (1988). (Received 13 December 1989; accepted 17April 1990)