Lecture 17c Lambert Conformal Conic Projection PDF

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GMAT2700 Lecture 17c Introduction to Lambert Conformal Conic Projection

• Basic concept of Lambert Conformal Conic (LCC) Projection • How are the mapping equations created? • Examples of LCC projection

GMAT2700, Lecture 17c: Lambert Conformal Conic Projection J. Wang, School of Civil and Environmental Engineering, UNSW

Conic Projection: The concept Conic projection can be visualised as the projection of an ellipsoid onto a cone. The apex of the cone is centered in the extension of the polar axis of the ellipsoid. Meridians appear as straight lines radiating from a point beyond the mapped area Map wrapped on a cone http://www.progonos.com/furuti/MapProj/Dither/ProjCon/projCon.html

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Features of Normal Conic Projections Compared with the sphere/ellipsoid, angular distance between meridians is always reduced by a fixed factor, the cone constant.  Parallels are arcs of circles, concentric in the point of convergence of meridians.  Parallels cross all meridians at right angles.  Distortion is constant along each parallel

Cone Flattened onto a Plane

Secant vs. Tangent Conical Projections

Secant Conical Projection This projection employs a cone intersecting the ellipsoid at two parallels known as the upper and lower standard parallels for the section to be presented.

Tangent Conical Projection This projection employs a standard parallel that represents a line of tangency between the cone and the surface of the ellipsoid.

Secant Condition of LCC

(See also the notes for previous Lectures)

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Introduction to Lambert Conformal Conic (LCC) Projection Since LLC is conformal projection, distortion is comparable to that of the Transverse Mercator projection. Distances are true along the standard parallels. Directions are fairly accurate over the entire projection zone.

Lambert Conformal Conical Projection

Introduction to Lambert Conformal Conic (LCC) Projection Shapes usually remain relative to scale but the distortion increases away from the standard parallels. Shapes on large-scale maps of small areas are essentially true. Scale factor is exactly 1 at the standard parallel. Scale factor decreases between and increase away from the standard parallels.

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Brief History of LCC For mapping an area of considerable extent in longitude, a geometrical system based on the notion of an east-west centre line can be used  Johann Heinrich Lambert devised and published LCC in 1772  LCC projection remained almost unknown untill the begining of the World War I  LCC was introduced and has been brought to conspicuous attention by the French Military Survey under the name “Quadrillange kilométrique systéme Lambert” (Tardi, 1934)  Quick computations of distances are possible  Determination of azimuth of lines joining any two points within artillery range are of great value to military operation. Since 1947 the LCC has been superseded by the UTM for world-wide military mapping.

Johann Heinrich Lambert (1728-1777) Inventor of the TM, the Conformal Conic, the Azimuthal Equal-Area, and other important projections,

LCC and its Grid System Projected point Point to be projected

Hooijberg, 1997, page135

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Lambert’s Conical Zone: An Example

Hooijberg, 1997, page138

LCC Mapping Equations

Symbols and Definitions

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LCC Mapping Equations Major reference: Richardus P. & Adler P. K. (1972): Map Projections

One Standard Parallel Case The coordinates of an arbitrary point P are given in a polar coordinate system (R, γ): X  Rb  R cos   Y  R sin  

γ

R

Rb

(1)

For the derivations on the ellipsoid, the distance element is taken as: ds2   2 d2  2 cos2  d2

(2)

with, e   2 and g   2 cos 2 

On the conical surface this element is given: dS2  dR2  R2 d 2

Development of conical projection surface

(3)

with, E  1 and G  R 2

LCC Mapping Equations Equation (3) can be proven as follows:

To derive mapping equations, conditions are set up: (1) R is a function of φ only: R  q1( ), (2) γ is a function of λ:

  c1  c2 ,

R 0    0, 

  c1 

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LCC Mapping Equations Using such conditions, the Gaussian quantities for projection (conical) surface are obtained as: 2   R  2               E            0  F           G 2            0     

  R 2 R     2         E   R       R R R F                   G   2  R  R    2       

   1   R   0  0        2 2 R        0        0

0

  R  2   R 2         2           R  , i.e) E    and G  c12R 2    0 0      2 2   2 2       R   c1 R        

LCC Mapping Equations The condition of conformality is: E G   k 2, e g

2

hence

c 2R 2 1  R    2 1 2  k2  2        cos 

(4)

From (4), we obtain: dR c d  1 R  cos 

(5)

Integrating (5) results in: e    1  esin   2   lnR  -c 1 ln tan(45  )    lnc 3 2  1  esin       so that,

   1  esin   R  c 3  tan(45  )  2  1  esin    

e 2

    

(6)

c1

where, e 

a2  b2 2 a

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LCC Mapping Equations The constants c1, c2 and c3 are to be determined: first the c3: in the origin O(φ0, λ0), the cone being tangent to the ellipsoid: R  Rb   0 cot 0

(7)

With equation (6) it is seen that e      1 esin  0  2 R b   cot 0  c 3  tan(45   0 )   2  1 esin  0      Therefore,  cot 0 c3  c1 e     0  1 e sin 0 2     tan(45  ) 2  1 e sin 0    

c1

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LCC Mapping Equations According to equation (4), the scale factor k: k

c 1R

(8)

 cos 

As k0=1, and the variation of k should be minimum radiating from the origin, the condition should be satisfied that (ək/əφ)0 is equal to zero. Differentiating (8) with respect to φ gives at the origin:  0 sin 0  R   k  1     0  c 1   ( 0cos 0)  c 1R b  2cos 2  0  0  0 0 Now from equation (5)  R  c 1R b 0   k 0 0        0  0 cos  0

(9) k0  1

So, equation (9) becomes, since k0 = 1:

c1

cR 0 0 sin 0  1 b 0 cos 0 0 cos 0 0 cos 0

so that, c 1  sin  0

LCC Mapping Equations The constant c2 is equal to zero, if at the origin: 0

if

 0

The transformation formulae of the LLC projection thus become: e    tan(45   )1  esin  2     2 1  esin    R  Rb  e  1  esin  0  2    0    tan(45  ) 2 1  esin  0   

sin 0

(10)

   sin  0 1

k

Rsin 0 (1  e2 sin 2  )2 R sin  0  a cos   cos

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LCC Mapping Equations Two Standard Parallels Suppose the area to be mapped fits between the two parallel circles of latitude φ1 and φ2 and that at these limits the scale distortion should be the same. Namely, R 1 sin  0 R 2 sin  0 k1 

 1 cos  1

 k2 

 2 cos  2

(11)

or R1 v cos 1  1 R2 v 2 cos 2

(12)

or  1  1  e   tan(45  )  2 1 e  v 1 cos 1   v2 cos 2   tan(45 1 )  1  e   2  1  e  

e sin 1  2    sin 1   

sin  0

(13)

e  sin 0 sin  2  2  

 sin  2   

Richardus P. & Adler P. K. (1972): Map Projections

LCC Mapping Equations The value of sin φ0 can be determined from equation (13) as: sin 0 

ln  1 cos 1  ln  2 cos 2 1/ 2 e 1/ 2e    1 e sin 1   1  e sin 2   (14)   ln tan(45 1 / 21 )   ln  tan(45 1 / 21 )    1 e sin  1   1  e sin 2    

In the conical projection, the scale factors have unity at two parallels, so that: R 1 sin  0

 cos1 1



R  sin  0 1   cos 2 2

2

or R  sin  0   1 cos 1  1  R  sin 0  2 cos 2  2 Richardus P. & Adler P. K. (1972): Map Projections

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LCC Mapping Equations  Two Standard Parallels e   tan(45  1)  1 e sin  1  2     2  1 e sin  1   R1  R b e  1 sin  0  2  tan(45   0 )   e   2  1 e sin  0  Rb  C e 02

  1  e sin   tan(45  0 ) 2  1  e sin  0  



sin  0

       

 1 cos  1

e     1  e sin 1  2    sin 0 tan(45  1 ) 2  1  e sin 1     

sin0

 2 cos 2 e   2  1  e sin  2  2    sin  0 tan(45  )  2  1  e sin  2     

sin 0

LCC Mapping Equations  Two Standard Parallels e    1  e sin   2      R  C tan(45  ) 2  1 e sin          sin  0 R sin  0 k  cos

sin 0

In conical projections, the scale errors vary increasingly with the range of latitude north or south of the standard parallels Richardus P. & Adler P. K. (1972): Map Projections

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LCC Scale Factor  A reference meridian and standard parallel are assumed with: - Tangent along the standard parallel (k =1) or - Secants along two standard parallels (k = 1)

 On two selected standard parallels, Arcs of longitude are usually represented to a scale factor k < 1

 Between two parallels, the scale factor is k < 1  Outside two parallels, the scale factor k < 1

LCC Mapping Equations (Hooijberg, 1997)  Computation of Projection Zone and Ellipsoid Constants Q1 

1  1 sin  1 1 esin 1   e ln  ln  1 esin 1  2  1 sin  1

W1  1- e2 sin 2 1 Sim ilarly for Q u ,W u , Q b ,Q o and Wo upon s ubs titution of the appropriate latitude. sin  o  K Rb ko

lnWu cos  1/(W1 cos  u )  Q u Q 1

  acos 1 eQ 1 sin o a cos  u eQu sin o  W1 sin o Wu sin o K K Ro  Q b sin o ,  Qo sin o e e W tan o Ro , No  o  Rb  Nb  Ro a



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LCC Mapping Equations (Hooijberg, 1997) Direct Conversion Computation Input: Geographical coordinate of point P (φ, λ) on meridian API, projected on Spi Output: Grid coordinate of point p (E, N), convergence angle (γ), grid scale factor (k)

1  1  sin  1  e sin   e ln  ln 2  1  sin  1  e sin   K R  Qsin o e E  E0  R sin N  R b N b  R cos  Q

  (0   ) sin 0 1

k  (1- e 2 sin 2  ) 2 (R sin0 ) /(a cos  )

LCC Mapping Equations (Hooijberg, 1997) Inverse Conversion Computation Input: Grid coordinate of point p (E, N) on a line Spi Output: Geographical coordinate of point P (φ, λ), convergence angle (γ), grid scale factor (k)

R  Rb N  Nb E   E - E0 E R   0   / sin 0

1   tan 

2 2 R  R  E ln (K/R) Q sin0

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LCC Mapping Equations (Hooijberg, 1997) Inverse Conversion Computation Input: Grid coordinate of point p (E, N) on a line Spi Output: Geographical coordinate of point P (φ, λ), convergence angle (γ), grid scale factor (k) Us e an approxim ation for  as follows e2Q 1 and iterate sin  as follows : e2Q 1 1  1 sin  1 esin   f 1  ln  ln  Q 2  1 sin  1 esin  

sin  

1 e2  2 2 1- sin  1- e sin 2  sin  sin  ( f1 /f2 ) and iterate to obtain  with sufficient accuracy

f2 

1

k  (1- e 2 sin2  ) 2 (R sin 0 ) /(a cos )

Lambert’s Conformal Conical Projection Applications Reference and LCC-Projection Systems of France - The Lambert’s conformal IGN-grid with two standard parallels is a Lambert grid with one standard parallels to which a grid scale constant has been applied (IGN, 1994). - The longitude of the Meridians of Paris, 2g. 59 68 98 E of Greenwich, is used for the datum for National Surveys and Maps. - Geodetic coordinates – expressed in centesimal units- are used for the civilian surveys in France since 1920

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Lambert Conical Conformal Projection of USA

Oklahoma Lambert Projection: North and South Zones

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RTA Lambert Projection - Dennis Entriken

RTA Lambert Projection - Dennis Entriken

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RTA Lambert Projection - Dennis Entriken

RTA Lambert Projection - Dennis Entriken

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RTA Lambert Projection - Dennis Entriken

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Things to do 

Read the lecture notes



Use the Excel Spreadsheet for Zone to Zone transformation provided at the following website http://www.icsm.gov.au/icsm/gda/gdatm/index.html



Preparation for class discussions in week 12 (see the instructions at the course website).

LCC Mapping Equations Equation (3) can be proven as follows: dS 2 dX 2 dY 2 X X dX  dR d   -cos  dR  Rsin  d  R  Y Y dX  dR d  s in dR  Rcos d R  dS 2  (cos 2  s in2 ) dR2  R dR d cos dR  R dR d cos dR  (sin2  cos 2 )R 2d 2  dR2  R2 d 2

To derive mapping equations, conditions are set up: (1) R is a function of φ only: R  q1(), (2) γ is a function of λ:

  c1  c2 ,

R 0    0, 

  c1 

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