5.7 Volume Conserving Affine Transformation 73
5.7 Volume Conserving Affine Transformation
5.7.1 Mapping of a Sphere onto an Ellipsoid
LetrandrAbe the coordinates in the original and in an affine transformed space.
The components are linked by
rμArμA=rμAμνrν, rAμ=A^1 μν/^2 rν. (5.53)When the tensorAμνhas positive eigenvalues, the relations (5.53) describe a mapping
of a sphererA·rA=const. onto an ellipsoid inr-space. The affine transformation
matrix, as considered in Sect.2.3.2,isA
1 / 2
μν. The volume of the ellipsoid is equal to
that of the sphere provided that the product of the eigenvaluesAi,i= 1 , 2 ,3ofAμν
are equal to 1, viz.,
A 1 A 2 A 3 = 1. (5.54)
For a uniaxial ellipsoid two of the eigenvalues are equal, e.g.A 2 =A 3.
5.7.2 Uniaxial Ellipsoid
Letube a unit vector parallel to the symmetry axis of a uniaxial ellipsoid. In this case,
one can make the ansatzAμν∼δμν+Auμuνwhere thenon-sphericity parameter
Ais bounded according to− 3 / 2 <A<3. The volume conserving condition (5.54)
implies
Aμν=[(
1 −
1
3
A
) 2 (
1 +
2
3
A
)]− 1 / (^3) (
δμν+Auμuν
)
. (5.55)
The equation (5.54) withrμArμA=rA^2 =const. describes an ellipsoid with the semi-
axesa=[( 1 −^13 A)( 1 +^23 A)−^1 ]^1 /^3 rAandb=c=[( 1 +^23 A)( 1 −^13 A)−^1 ]^1 /^6 rA.
Thus the axes ratio is
Q=
a
b=
[(
1 −
1
3
A
)(
1 +
2
3
A
)− 1 ]^1 /^2
, (5.56)
andAis related toQby
A= 3