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∂U M
∂u μ ,
F l
F r
U A
versus u α
versus
∂u .
(1.71)
∂U
d U 1 , d U 2
Indeed, they are generated from a dual holonomic base (coordinate base)
{
}
versus
d u 1 , d u 2
d V 1 , d V 2
d v 1 , d v 2
{
}
to an anholonomic base
{
}
=
{
Ω 1 2 }
versus
{
}
=
{
ω 1 2 }
by the
transformations
d U 1
d U 2 =F l Ω 1
d u 1
d u 2 =F r ω 1
.
versus
(1.72)
Ω 2
ω 2
E λ 1 2 , right Tissot circle
S 1 , the tangent vectors are
Fig. 1.8. Right Cauchy-Green tensor, right Tissot ellipse
∂/∂u and ∂/∂v
1-3 Two Examples: Pseudo-Cylindrical and Orthogonal Map
Projections
Two examples of deformation analysis: pseudo-cylindrical and orthogonal map projections
(Cauchy-Green deformation tensor, its eigenspace, Tissot ellipses of distortion).
The general eigenspace analysis of the Cauchy-Green deformation tensor visualized by the Tissot
ellipses of distortion is the heart of any map projection. It is for this reason that we present to
you the pseudo-cylindrical map projection called Eckert II as Example 1.5 and the orthogonal
projection of the northern hemisphere onto the equatorial plane as Example 1.6 . We recommend
to go through all details with “paper and pencil”.
Example 1.5 (Pseudo-cylindric map projection of type Eckert II, left Cauchy—Green deforma-
tion tensor).
Eckert ( 1906 ) proposed six new pseudo-cylindrical map projections of the sphere which have some
intrinsic properties. (i) The images of the central meridian and the pole have half the length of
the equator, the line of zero latitude. (ii) The images of lines of equilatitude, called parallel circles ,
are parallel straight lines. Consult Fig. 1.9 for a more illustrative information. For instance, as a
special pseudo-cylindrical projection, an equiareal mapping of the sphere onto a cylinder of type
 
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