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Hintergrund: Die Inversion von Karlheinz Wagners Kartennetzentwürfen

Background: The Wagner Projections Inversion

With best regards to Adrian Weber, Institut für Kartographie, ETH Zürich.

Direct formulas are the common map projection formulas and transformes the spheric Earth surface coordinates λ (longitude) and φ (latitude) into the map coordinates x and y. The inverse formulas transformes a map-x,y-pair into λ and φ.

Here the complete Wagner 1 to 9 Formulas - and their inversions.

Wagner I

Direct formula (published by Karlheinz Wagner 1949 p. 181) x = ( ( 2 * cfr3 ) / 3 ) * λ * cos(ψ) (1)

(Printing error: The 2 is forgotten in the Wagner 1949 manuscript)

y = cfr3 * ψ (2)

sin(ψ) = ( cr3 / 2 ) * sin(φ) (3)

(cr3 and cfr3 are constants - see below in the appendix)

Inverse formula

First compute the ψ. The (2) inversion is:

ψ = y / cfr3 (4)

(3) with the ψ from (4) gives:

φ = arcsin ( ( 2 / cr3 ) * sin(ψ) ) (5)

Note: |ψ| must be < π / 2 (else mirrored Earth images).

(1) with the ψ from (4) gives:

λ = ( x * ( 3 / ( 2 * cfr3 ) ) ) / cos(ψ) (6)

Wagner II

Direct Formula (Wagner 1949 p. 187)

x = c0.92483 * λ * cos(ψ) (7)

(To a better comparision with the original Wagner text I wrote as Wagners „0,92483“ a „c0.92483“. Take that „c0.92483“ as a constant name and see below in the appendix a more precise value.)

y = c1.38725 * ψ (8)

sin(ψ) = c0.88022 * sin(c0.8855*φ) (9)

Inverse Formula

First compute the ψ:

ψ = y / c1.38725 (10)

Note: If |ψ| > π / 2 than it gives a false mirrored Earth image.

Than the λ and φ:

λ = x / ( cos(ψ) * c0.92483 ) (11)

φ = ( 1 / c0.8855 ) * arcsin ( sin(ψ) / c0.88022 ) (12)

Wagner III

Direct Formula (Wagner 1949 p. 190)

x = q * λ * cos( 2 * φ / 3 ) (13)

q is constant, computed by means of φ0, a standard parallel latitude:

q = cos(φ0) / cos( 2 * φ0 / 3 ) (13a)

y = φ (14)

Inverse Formula

λ = x   /   ( ( cos( (2/3) * φ ) ) * q ) (15)

φ = y (16)

Wagner IV

Direct Formula (Wagner 1949 p. 192)

2*ψ + sin(2*ψ) = c2.96042 * sin(φ) (17)

Solvable by means of Newton-Raphson iteration - more easy as commonly thinked (G I Evenden)

x = c0.86309 * λ * cos(ψ) (18)

y = c1.56547 * sin(ψ) (19)

Inverse Formula

First the ψ-computation:

ψ = arcsin( y / c1.56547 ) (20)

... and now the λ and φ. Note, that the inversion doesn't need the Newton-Raphson iteration:

λ = x / c0.86309 * cos(ψ) (21)

φ = arcsin ( ( 2*ψ + sin(2*ψ) )   /   c2.96042 ) (22)

Wagner V

Direct Formula (Wagner 1949 p. 196)

2*ψ + sin(2*ψ) = c3.00895 * sin(c0.8855 * φ) (23)

(See above: Newton-Raphson iteration ...)

x = c0.90977 * λ * cos(φ) (24)

y = c1.65014 * sin(ψ) (25)

Inverse Formula

Here the inversion also doesn't need the Newton-Raphson iteration ...

ψ = arcsin( y / c1.65014 ) (26)

λ = x   /   ( c0.90977 * cos(ψ) ) (27)

φ = ( 1 / c0.8855 )   *   arcsin ( ( 2*ψ + sin(2*ψ) )   /   c3.00895 ) (28)

Wagner VI

Direct Formula (Wagner 1949 p. 196)

sin(ψ) = (cr3/π) * φ (29)

x = λ * cos(ψ) (30)

y = (π/cr3) * sin(ψ) (31)

Inverse Formula

ψ = arcsin( (cr3/π) * y ) (32)

λ = x / cos(ψ) (33)

φ = (π/cr3) * sin(ψ) (34)

Wagner VII and Wagner VIII

(The Wagner VII is also known as Hammer-Wagner)

General transversal eqal-area azimuthal series: Usable for the Wagner VII and VIII, but also usable for the Transversal Lambert Azimuthal Projection and the Hammer Projection.

Constants genesis and directory

 CM CM2 CN CA CB Transversal Lambert 1 1 1 1 1 Hammer 1 1 1/2 2 2 Wagner VII Values genesis m 1 n 2k/sqrt(m*n) 2/(k*sqrt(m*n)) Wagner VIII Values genesis m1 m2 n 2*k/sqrt(m1*m2*n) 2/(k*sqrt(m1*m2*n)) Wagner VII(=Configuration 65-60-60-0-200) 0.9063077870366490 1 1/3 1/2 * 5.3344669029266510 2.4820727672498521 Wagner VIII(=Configuration 65-60-60-20-200) 0.9211662819778872 c0.8855 (See Wagner II) 1/3 1/2 * 5.6229621893185126 2.6163066666935492 Wagner VII, Canters Optimisation(Conf. 53.4560-125.5860-60-0-212.35) 0.8033998325209590 1 0.6977000000000000 1/2 * 3.7993481624866600 1.8782372027354400 Wagner VIII, Canters Optimisation(Conf. 55.4366-118.8540-60-20-196.54)) 0.8369998160404740 0.8855017059025980 0.6603000000000000 1/2 * 4.2026161570088900 1.9448414568445100 Wagner VIII Variation 1(Configuration 65-84-60-25-200)„Meta-Canters“ 0.9302347463364700 0.8552968757751740 0.4666666666666670 1/2 * 4.1594862782149100 2.5900256879460900 Wagner VIII Variation 2(Configuration 75-108-60-0-200)„Meta-TsNIIGAiK“ 0.9659258262890680 1 0.6 1/2 * 3.2228657925102900 2.1415229944645200 Wagner VIII Variation 3(Configuration 57-105-60-20-200)„Batwing“ 0.8524201821156020 0.8855017059025980 0.5833333333333333 1/2 * 3.3056978633041300 2.7481277305053100 Wagner VIII Variation 4(Configuration 60-132-60-0-200)„Eurasia Map“ 0.8660254037844380 1 0.7333333333333333 1/2 * 2.6257228616347200 2.3987171932747300

Remark I: The 5 Configuration values are: pole line length latitude ψ1, „Umbezifferung“ longitude λ1, area distortion reference latitude φ1 (always 60°), area distortion percent in the area distortion reference latitude S60, equator-meridian ratio percent p.

Remark II: The (not in the formulas needed) k-Values are -
Wagner VII: 1.4660144724344600,
Wagner VIII: 1.4660144724344600,
Wagner VII/Canters Opt.: 1.4222610848373400,
Wagner VIII/Canters Opt.: 1.4700014433629400,
Wagner VIII V1: 1.2672660848659700,
Wagner VIII V2: 1.226760374592110,
Wagner VIII V3: 1.0967638345702100,
Wagner VIII V4: 1.0462486695360900.

Direct Formula (Wagner p. 206 (Wagner VII) and 209 (Wagner VIII))

sin(ψ) = CM * sin(φ * CM2) (35) (Korrigiert nach freundlichem Hinweis von Tiha von Ghyczy, 12.10.2012, vielen Dank)

cos(δ) = cos( CN * λ ) * cos(ψ) (36)

cos(α) = sin(ψ) / sin(δ) (37)

x = 2 * CA * sin(δ/2) * sin(α) (38)

y = 2 * CB * sin(δ/2) * cos(α) (39)

Inverse Formula

Explanation:

We begin with a (39) transformation:

2 * sin(δ/2) = y   /   ( CB * cos(α) ) (40)

Now we set (38) equal (39), give (40) into (38):

x = CZ * ( y / (CB * cos(α) ) * sin(α) (41)

A sine in the enumerator and a cosine in the denominator - that is a tangent ...

x = CA * ( y / CB ) * tan(α) (42)

tan(α) = ( CB * x ) / ( CA * y ) (43)

Formula:

α = arctan(( CB * x ) / ( CA * y )) (44)

With the y-formula (39):

y = 2 * CB * sin(δ/2) * cos(α) (45)

we can compute now δ:

δ = 2 * arcsin ( y / ( CB * cos(α) ) ) (46)

Now we have α and δ an can go on with ψ by means of an (37) inversion:

ψ=arcsin( cos(α) * sin(δ) ) (47)

The inverse (35) can now give the latitude ...

φ = arcsin( sin(ψ) / CM ) * (1 / CM2) (48)     (... the * (1 / CM2) is my Original manuscript typography)

... and the inverse (36) gives the longitude:

λ = (1 / CN) * arccos ( cos(δ) / cos(ψ) ) (49)

That's the inverse Hammer, Wagner VII and Wagner VIII.

Wagner IX

(or Wagner-Aîtoff)

Direct Formula (Wagner 1949 p. 215)

ψ = (7/9) * φ (50)

cos(δ) = cos( λ * (5/18) ) * cos ( φ * (7/9) ) (51)

cos(α) = sin( φ * (7/9) ) / sin(δ) (52)

x = c3.6 * δ * sin(α) (53)

How does Wagner calculate his projections?
Today unthinkable - only by means of slide rule and log table. It is possible that Wagner does't recognise his constant 3,60 as exactly 3.6:
c3.6 = k/sqrt(m*n) = (sqrt(14/5))/(sqrt((7/9)*(5/18))) = sqrt((14*9*18)/(5*7*5)) = sqrt(2268/175) = 3.60000000000000000000000000000000

y = c1.28571 * δ * cos(α) (54)

It seems that Wagner does't recognise his constant 1,285714 as 9/7:
c1.2571 = 1/(k*sqrt(m*n)) = 1/(sqrt(14/5)*sqrt((7/9)*(5/18))) = sqrt((14*7*5)/(5*9*18)) = sqrt(490/810) = 9/7

Inverse Formula

Explanation:

Separate the δ in (53) and (54)

δ = x / ( c3.6 * sin(α) ) (55)

δ = y / ( c1.28571 * cos(α) ) (56)

Set (55) equal (56):

x / ( c3.6 * sin(α) ) = y / ( c1.28571 * cos(α) ) (57)

and separate the α:

cos(α) / sin(α) = (y * c3.6) / (x * c1.28571) (58)

That is the contangens of α:

cot(α) = (y * c3.6) / (x * c1.28571) (59)

Formula:

α = arccot( (y * c3.6) / (x * c1.28571) ) (60)

Note: if y < 0 and x > 0 add π to α and if y < 0 and x < 0 then sub π from α

Now δ is computable by means of (55) or (56):

δ = x / ( c3.6 * sin(α) ) (55)

The inversion of (52) gives now φ:

φ = (9/7) * arcsin( cos(α) * sin(δ) ) (61)

And the (51) inversion gives λ:

λ = (18/5) * arccos( cos(δ) / cos( (7/9) * φ ) ) (62)

That's the inverse Wagner-Aîtioff

Appendix: Variables and Constants

Variables

λ ... longitude

φ ... latitude

ψ ... reduced longitude

φ0 ... standard parallel longitude

α ... plane polar angle

δ ... plane polar distance

x ... plane map coordinate 1

y ... plane map coordinate 2

q ... a Wagner III constant

Some roots of 3

cr3 = 1.7320508075688772935274463415059 („root of 3“)

cfr3 = 1.316074012952492460819218901797 („fourth root of 3“)

Wagner II constants

c0.92483 = 0.92483273372222111597803131070463

c1.38725 = 1.3872491005833316739670469660569

c0.88022 =0.88022348777441292844983304540509

c0.8855 = 0.88550170590259964505240645734844 (also used in other projections)

The 79 degree value

c79.695 = 79.69515353123396805471658116136 (c0.8855 * 90. - Not here used, but often in the Wagner manuscript)

Wagner IV constants

c0.86309 = 0.86309513988625768962483088739305

c1.56547 = 1.565481415999337518303982239654

c2.96042 = 2.9604205061776341390721520929393

c0.86309 = 0.86309513988625768962483088739305

Wagner V constants

c0.90977 = 0.9097725087960359780692854133276

c1.65014 = 1.6501447980520194242829775328104

c3.00895 = 3.0089552244534209263760071797089

Wagner VII and VIII constants

0.9063... = 0.90630778703664996324255265675432 (sin(65°))

1.4660... = 1.4660144724344624720643223450521 (Not here used, but in Wagners text. It is Wagners k variable)

5.3344... = 5.3344669029266510740930291868717

2.4820... = 2.4820727672498521169179809709757

0.9211... = 0.92116628197788727359146199441524

5.6229... = 5.622962189318512646048805432804

2.6163... = 2.6163066666935492312865258611159 (Wagner: 2.6162)

Wagner IX constants

c3.6 = 3.60000000000000000000000000000 (sqrt(2268/175))

c1.28571 = 1.2857142857142857142857142857143 (9/7)

Comment

k, n, m, m1, m2 are variables in the original Wagner - here only written as „comment variables“. If you don't compare my formulas with Wagners book - ignore it.

Last but not least

π = 3.1415926535897932384626433832795

Literature

Böhm, R.: Variationen von Weltkartennetzen der Wagner-Hammer-Aîtoff-Entwurfsfamilie. Kartographische Nachrichten 56. Jg., Nr. 1., p. 8-16. - Kirschbaum: Bonn 2006.
Canters, F.: Small-scale Map Projection Design (p. 185). London: Taylor & Francis 2002.
Wagner, K.: Kartographische Netzentwürfe. Leipzig: Bibliographisches Instuitut 1949.

Durchsicht 12.10.2012