Strain Analysis of Common Map Projections in Iran Using...

Geomatics 93 (National Conference) – May 2014

Strain Analysis of Common Map Projections in Iran Using Continuum

Mechanics Concepts

A. Rastbood

Assistant Professor, Surveying Engineering Department, Tabriz University, Tabriz, Iran

arastbood @tabrizu.ac.ir

ABSTRACT

Some of the most commonly used map projections, are analyzed from

the viewpoint of the deformation introduced, when the objective surface

is mapped into a plane. The analysis of the strain tensor structure is

carried out concerning map projections with particular interest on

invariant scalar strain parameters, i.e., dilatation and maximum shear

strain. After a short theoretical account, graphs for the computation of

these parameters are given for common projection systems in Iran.

These projection systems include UTM and Lambert conic map and

other projections . The derived parameters of these projection systems

may be used as criteria for the choice of a convenient map projection in

the interested region in addition to the traditional criteria of

conformality, equivalence, etc. The theory of this subject has been

developed by A. DERMANIS and E. LIVIERATOS and in this paper we

only apply it to the map projections used in the region of Iran.

1- Introduction

Applications of the Theory of Elasticity in Cartography appear in the literature since

the late 19th Century (Tissot 1881, Fiorini 1881). Interesting work has been carried out

up to now, utilizing in abstraction the tools of mechanics for the study of deformations

induced when original figures on a Riemmanian space, (sphere or ellipsoid) considered as

the objective space, are mapped by a one-to-one correspondance on a Euclidean space.

In this paper we are concerned with the representation of the invariant criteria of

dilatation and maximum shearing strain, on some well known map projections in the

region of Iran. This is done using the local strain tensors derived by mapping

differentially the local azimuthal plane onto the plane of representation. The analysis

follows the Lagrangian approach, thus the deformation maps are referred to the objective

surface of a sphere [1, 4].

2- Strain Tensor

One-to-one correspondance of a plane space with another one is locally represented

by the differential mapping


( )( )

∂

∂

∂

∂∂

∂

∂

∂

=∂

∂=

2

2

1

2

2

1

1

1

21

21

,

,

α

β

α

βα

β

α

β

αα

ββG (1)

where ( )T

21,ααα = and ( )T

21, βββ = the respective sets of orthogonal cartesian

coordinates, the relevant metrics being

( )ααββ

ββαα

β

α

GdGdddds

dGGdddds

TTT

TTT

==

== −−

2

112

(2)

It is hereon assumed that ( ) 0det ≠G at the points of interest.

In the Lagrangian approach the strain tensor LE is a function of the objective surface,

( )αLE , defined by the relation

αααβ dEddsds L

Tdef

222 =− (3)

and consequently

( )IGGET

L −=2

1 (4)

where I the unit matrix

Dealing with the mapping of a curved space onto another curved space, we can pass

to the corresponding local tangent spaces and operate as above, since the strains involved

are differential quantities.

3- Cartographic Application

Considering the mapping (Fig 1.)

Fig 1. Mapping

where ( )ϕλ, orthogonal spherical coordinates on a unit sphere, ( )yx, cartesian

coordinates on the map plane and ( )ϕλ ss , coordinates on the local tangent plane, i.e., λs

and ϕs are lengths along the local parallel and meridian respectively. The corresponding

differential mappings are (Fig 2.)


Fig 2. differential mappings

where 1−= JQS (5)

and

( )( )

( )( )

( )( )

.,

,,

,

,,

,

,

ϕλϕλ

ϕλ

ϕλ ss

yxQ

ssQ

yxJ

∂

∂=

∂

∂=

∂

∂= (6)

From the well known relations

λϕ

ϕ

λ

φ

dds

dds

cos=

= (7)

It follows that

10

0cos

1

ϕ=Q (8)

Q being diagonal due to the orthogonality of the geographical grid on the sphere.

3-1- Lagrangian Cartographic Strain

The mapping ( ) ( )dydxdsds ,, →φλ involves the Jacobin S, which introduced into (5)

provides the Lagrangian strain of map projections

( )ISSET

L −=2

1 (9)

or

( )ITEL −=2

1 (10)

where

SSTT= (11)

Replacing (5) into (11) we obtain

JQJQT TT= (12)

or

=e

f

fg

T

ϕ

ϕϕ

cos

coscos2

(13)

g, f, e being the Gauss forms defined by


JJef

fgT

def

=

. (14)

with 22

∂

∂+

∂

∂=

λλ

yxg

ϕλϕλ ∂

∂

∂

∂+

∂

∂

∂

∂=

yyxxf (15)

22

∂

∂+

∂

∂=

ϕϕ

yxe

The matrix is nothing but the Tissot tensor, from which the Tissot ellipse is

computed. Comparing (13) with (10) we observe that the Tissot tensor is not a pure strain

tensor but only a part of the Lagrangian strain tensor.

From (13) the matrix and minimum semi-axes of the Tissot ellipse, Ta and Tb , are

computed, as well as the direction Tψ of Ta

( ) ( )

ϕ

ϕψ

ϕ

ϕϕ

ϕ

2

2

22

22

2

cos

cos2arctan

2

1

cos

cos2cos

cos2

1

eg

f

fege

g

b

a

T

T

T

−

−=

+−

−

++=

(16)

By definition

TTT

TTT

b

a

γ

γ

−∆=

+∆=2

2

2

2 (17)

where T∆ and Tγ could be called the pseudo-dilatation and the pseudo-maximum shear

strain respectively, or the Tissot dilatation and the Tissot maximum shear strain.

The Lagrangian strain ellipse is computed from (10)

LLL

LLL

b

a

γ

γ

−∆=

+∆=

2

2 (18)

where

( ) ( )ϕ

ϕϕγ

ϕ

2

222

2

cos2

cos2cos

1cos2

1

feg

eg

L

L

+−=

−

+=∆

(19)

and the direction of La

TL ψψ = (20)

Comparing the above expressions we obtain the relations


TL

TL

γγ2

1

12

1

=

−∆=∆

(21)

Note that the real dilatation L∆ and the maximum shear strain Lγ are systematically

smaller than the ones referred to the Tissot tensor. The Lagrangian strain analysis

described above is referred to the sphere [3].

L∆ and Lγ in (18) as defined in (19) are invariant unitless scalar quantities, called in

continuum mechanics the dilatation and maximum shear strain respectively.

Dilatation represents the isotropic part of deformation. L∆ is called dilatation

following the current terminology in continuum mechanics. However it does not exactly

correspond to change of area per unit area (true dilatation), but only approximately for

infinitesimal deformations.

Maximum shear strain represents the anisotropic part of deformation and it is the

shear across the direction of its maximum value (always positive). Its significance is

alteration in shape independently of magnification or reduction.

From the well-known conformality conditions concerning the Gauss differential

forms

ϕ2cos

0

eg

f

=

= (22)

it follows that

ϕ

γ2cos

0

eL

L

=∆

= (23)

for conformal projections.

It is also known that if an area element dA on the unit sphere is mapped onto an

element Ad ′ on the plane, then

ϕcos

2fge

dA

Ad −=

′ (24)

For conformal projections conditions (22) give

edA

Ad=

′ (25)

The change of area per unit area (true dilatation) is

LedA

dAAd∆=−=

−′1 (26)

It follows that for conformal projections, L∆ corresponds exactly to true dilatation.

Perhaps the use of the term dilatation for the isotropic part of deformation is

somewhat misleading in the case of map deformation. L∆ is called dilatation within the

Theory of Linear Elasticity, where for infinitesimal deformations it is a first-order

approximation to true dilatation. Here we call dilatation the isotropic part of deformation

for a conceptual linkage, with the terminology in Elasticity.

L∆ and Lγ are obviously related to the semiaxes Ta , Tb of the Tissot indicatrix. In fact

it can be shown (DermanJs and Livieratos, 1981a) that


12

22

−+

=∆ TTL

ba (27)

2

22

TTL

ba −=γ (28)

4- Experiment Results

For experiments we selected some well known map projections and calculated

dilatation and maximum shear strain for the region of Iran in them using MATLAB

software. For this purpose first Gauss forms i.e. g, f, e being evaluated using

transformation formulas and equation (15) and then L∆ and Lγ are evaluated using

equation (19).

4-1- Lambert conformal conic projection For the Lambert conformal conic projection as normally used, with two standard parallels, the

equations for sphere may be written as follows: [5] Given ϕλϕϕϕ ,,,,, 0021R and λ :

θρρ

θρ

cos

sin

0 −=

=

y

x (29)

where

( )( )

( )

( )( ) ( ) ( )[ ]24tan24tanlncoscosln

24tancos

24tan

24tan

1221

11

00

0

ϕπϕπϕϕ

ϕπϕ

ϕπρ

λλθ

ϕπρ

++=

+=

+=

−=

+=

n

nF

RF

n

RF

n

n

n

(30)

00 , λϕ : the latitude and longitude for the origin of the rectangular coordinates

21, ϕϕ : standard parallels

Fig 3. Dilatation in Iran for Lambert conformal conic projection


Calculations showed that 0=Lγ and L∆ is only a function of ϕ . Figure 3 shows the

situation of L∆ in Iran for Lambert conformal conic projection with the following

characteristics: oooo 54,24,36,30 0021 ==== λϕϕϕ

4-2- Mercator's projection

For the sphere, the formulas for rectangular coordinates are as follows: [5]

( )( )24tanln

0

ϕπ

λλ

+=

−=

Ry

Rx (35)

or

( ) ( ) ( )( )[ ]ϕϕ sin1sin1ln2 −+= Ry

where R is the radius of the sphere at the scale of the map as drawn, andϕ and λ are

given in radians.

Fig 4. Dilatation in Iran for Mercator's projection

Calculations showed that 0=Lγ and L∆ is only a function of ϕ . Figure 4 shows the

situation of L∆ in Iran for Mercator's projection.

4-2-1- Transverse Mercator's projection

A partially geometric construction of the Transverse Mercator for the sphere involves

constructing a regular Mercator projection and using a transforming map to convert

meridians and parallels on one sphere to equivalent meridians and parallels on a sphere

rotated to place the equator of one along the chosen central meridian of the other. Such a

transforming map may be the equatorial aspect of the Stereographic or other azimuthal

projection, drawn twice to the same scale on transparencies. The transparencies may then

be superimposed at o90 angles and the points compared.

In the age of computers, it is much more satisfactory to use mathematical formulas. The

rectangular coordinates for the Transverse Mercator applied to the sphere: [5]

( ) ( )[ ]BBRkx −+= 11ln21 0

or

( )[ ]{ }000

0

costanarctan

arctan

ϕλλϕ −−=

=

Rky

hBRkx (36)


where

( )0sincos λλϕ −=B (37)

We calculated L∆ and Lγ for Transverse Mercator's projection in two situation with o00 =λ (Figure 5) and with o540 =λ (Figure 6).

Fig 5. Dilatation in Iran for Universal Transverse Mercator's projection

Fig 6. Dilatation in Iran for Universal Transverse Mercator's projection

Calculations showed that 0=Lγ and L∆ is only a function of both ϕ and λ . Figures 5

and 6 show the situation of L∆ in Iran for Transverse Mercator's projection.

4-2-2- Universal Transverse Mercator projection

We calculated L∆ and Lγ for Universal Transverse Mercator's projection in two

situation with 00 =k (Figure 7) and with 9996.00 =k (Figure 8) in zone 38S.

Calculations showed that 0=Lγ and L∆ is a function of both ϕ and λ . Figures 7 and 8

show the situation of L∆ in Iran for Universal Transverse Mercator's projection.


Fig 7. Dilatation in Iran for Universal Transverse Mercator's projection with k0 = 0

Fig 8. Dilatation in Iran for Universal Transverse Mercator's projection with k0 = 0.9996

5- Conclusion

In this paper the study of the invariant strain parameters i.e. L∆ (dilatation) and Lγ

(maximum shear strain) has been carried out in their Lagrangian expression, for several

known map projections. These parameters represent the isotropic and anisotropic

behaviour of map deformation and allow deeper understanding of geometric alternations

of the map, even when the maps under comparison belong to the same projectional family

due to their traditional properties, i.e. conformality, equivalence, etc. This could be of

interest not only in Mathematical but also in Thematic Cartography.

For the family of conformal projections the anisotropic part of deformation vanishes

( )0=Lγ while the isotropic part L∆ corresponds exactly to the change of area per unit

area (true dilatation) and can be used as a criterion for the comparison between conformal

projections.

For the family of equal-area projections, L∆ does not vanish, while true dilatation does.

In this case the anisotropic part Lγ giving a measure of the "change of shape" is a

criterion for comparison between equal-area projection.

For map projections outside the above main two families, both the isotropic part L∆ and

the anisotropic part Lγ of deformation may serve as criteria for comparison. However Lγ

representing deformation of shape has a more obvious significance especially for maps,

where preservation of shape is an important requirement.


In addition to the comparison of different map projections, L∆ and Lγ may also be used

for the selection of the appropriate standard parallel(s) in any specific projection, i.e., a

conic projection depending on one or two such parallels. If the projection is to be used for

the region 21 λλλ ≤≤ and 21 ϕϕϕ ≤≤ , optimality criteria may be introduced in the form,

mincos2

1

2

1

=∆∫ ∫ λϕϕλ

λ

ϕ

ϕdd

p

L (41)

e.g., for the Lambert conformal conic projection ( )0=Lγ with one or two standard

parallels or in the form,

mincos2

1

2

1

=∫ ∫ λϕϕγλ

λ

ϕ

ϕdd

p

L (42)

e.g., for the Bonne equal-area projection ( )0=∆L , where p an integer, usually 1=p or

2=p [2]. The optimal values of the latitudes of the standard parallel(s) can be found

either analytically utilizing techniques from the calculus of variations which is quite a

difficult task, or numerically utilizing techniques from optimization theory.

Correction of all conformal projections e.g. Lambert in different countries by equation

(41) and equal area projections using equation (42) is recommended.

Generally speaking, the strain criteria in map projections compared to the traditional

global properties, of conformality, equivalence, etc., can be considered as intrinsic

deformation properties of maps.

References

1. DERMANIS A. and LIVIERATOS E., 1981a : Dilatation, Shear, Rotation and Energy

Analysis of Map Projections. Proceedings of the Vlll Hotine Symposium on

Mathematical Geodesy, Como, 7-9 September 1981, Boll. Geod. e Sci. Aff., Vol. XLII,

No. 1, 1983, pp. 63-68.

2. DERMANIS A. and LIVIERATOS E., 1981b: Strain Analysis of Map Projections.

Quaterniones Geod., Vol. 2, No 3, pp. 205-207.

3. DERMANIS A. and LIVIERATOS E., 1983: Applications of Deformation Analysis in

Geodesy and Geodynamics. Reviews of Geoph. and Space Phys. Vol. 21, No. 1, pp. 41-

50.

4. DERMANIS A. and LIVIERATOS E., 1983: Applications of Strain Criteria in

Cartography. Bulletin Geodesique, 57, 215-225.

5. SNYDER J. P., 1987: Map Projections – a Working Manual. U.S. Geological Survey

Professional Paper 1395. United States Government Printing Office, Washington.

Strain Analysis of Common Map Projections in Iran Using...

Documents

Transcript of Strain Analysis of Common Map Projections in Iran Using...