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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
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
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| 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)) |
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Wagner VII (=Configuration 65-60-60-20-200) |
0.9063077870366490 | 1 | 1/3 | 1/2 * 5.3344669029266510 | 2.4820727672498521 |
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Wagner VIII (=Configuration 65-60-60-20-200) |
0.9211662819778872 | c0.8855 (See Wagner II) | 1/3 | 1/2 * 5.6229621893185126 | 2.6163066666935492 |
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Wagner VII, Canters Optimisation (Conf. 53.4560-125.5860-60-0-212.35) |
0.8033998325209590 | 1 | 0.6977000000000000 | 1/2 * 3.7993481624866600 | 1.8782372027354400 |
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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 |
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Wagner VIII Variation 1 (Configuration 65-84-60-25-200) „Meta-Canters“ |
0.9302347463364700 | 0.8552968757751740 | 0.4666666666666670 | 1/2 * 4.1594862782149100 | 2.5900256879460900 |
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Wagner VIII Variation 2 (Configuration 75-108-60-0-200) „Meta-TsNIIGAiK“ |
0.9659258262890680 | 1 | 0.6 | 1/2 * 3.2228657925102900 | 2.1415229944645200 |
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Wagner VIII Variation 3 (Configuration 57-105-60-20-200) „Batwing“ |
0.8524201821156020 | 0.8855017059025980 | 0.5833333333333333 | 1/2 * 3.3056978633041300 | 2.7481277305053100 |
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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)
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 = 0.86309513988625768962483088739305c1.56547 = 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.