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To design a map with the dimensions of the Peters-Map using the formula for equal-area cylindrical projections of an ellipsoid the reference data of the Peters-Map must be known.

The reference ellipsoid of Friedrich Wilhelm Bessel 1841
semi-major axis a  

6377,397155 km
semi-minor axis b  

6356,07896325 km
flattening f (a - b) / a

0,0033427731144651
    1 / f

299,152818859504
numeric eccentricity e SQRT(1 - (1- f)^2)

0,081696830396505

The area of the ellipsoid zone between the equator and 1° N  Z = 4430108,29 km².
The area of the zone segment S = Z / 180 = 24611,71 km².
The length of the baseline B = SQRT(S) = 156,88120593 km.
The latitude of no distortion has the circumference U = Baseline * 180 = 28238,617067 km.
The radius of the latitude of no distortion r = U / 2 * Pi = 4494,315492 km.
The formula to compute the standard parallel Phi0 is:

Transformation to the standard circle with r = 1

a1 = 1
b1 = (1 / a) * b =  0,99665723
r1 = (1 / a) * r =  0,70472567

Phi0 = Arctan(SQRT(1 - r1^2) / (b1 * r1 ))

The standard parallel for the Peters-Map with ellipsoid of Bessel 1841 is Phi0 = 45,288534° N/S.

The value k for equal-area cylindrical projection on an ellipsoid::

        k = Cos(Phi0) / SQRT((1 - e^2 * Sin(Phi0)^2))

q = (1 - e^2) * (Sin(Phi) / (1 - e^2 * Sin(Phi)^2) - (1 / (2 * e)) * Log((1 - e * Sin(Phi)) / (1 + e * Sin(Phi))))
(Remark.: Log is the natural logarithms with the base e = 2.718281...)

this results in:

k at Phi0 = 45,288534° N/S k =   0,70472567
q at Phi = 90° N q90 =  1,99554446
y = a1 * q / 2 * k Height of the map 1 * (2 * q90) / 2 * k y =  2,83166137
x = a1 * Lambda * k Width of the map  1* 2 * Pi * k x =  4,42792198
The width-height ratio x / y XY = 1,56371875

For the table the following ellipsoid data have been used:

The example ellipsoid based on the
reference ellipsoid of Friedrich Wilhelm Bessel 1841
semi-major axis a  

1
semi-minor axis b  

0,996657226818421
flattening f (a - b) / a

0,0033427731144651
    1 / f

299,152818859504
numeric eccentricity e SQRT(1 - (1- f)^2)

0,081696830396505

All areas of the ellipsoid zones have been computed using the formula which Prof. Walter Buchholz derived for Arno Peters.

  Equal-area cylindrical projection of an ellipsoid with the standard parallel Phi0=45,288534° N/S)

Arno Peters method of construction
  k = 0,704725 Baseline = SQRT(0,00060514) = 0,02459957
           
phi (°) q y Ellipsoid zone Zone segment Zone segment / Baseline
0 0,000000 0,000000 0,000000 0,000000 0,000000
1 0,034672 0,024600 0,108925 0,000605 0,024600
2 0,069334 0,049192 0,217818 0,001210 0,049192
3 0,103975 0,073770 0,326646 0,001815 0,073770
4 0,138585 0,098325 0,435377 0,002419 0,098325
5 0,173154 0,122852 0,543979 0,003022 0,122852
6 0,207672 0,147342 0,652420 0,003625 0,147342
7 0,242128 0,171789 0,760667 0,004226 0,171789
8 0,276512 0,196184 0,868689 0,004826 0,196184
9 0,310815 0,220522 0,976453 0,005425 0,220522
10 0,345025 0,244794 1,083927 0,006022 0,244794
11 0,379132 0,268993 1,191079 0,006617 0,268993
12 0,413127 0,293112 1,297878 0,007210 0,293112
13 0,447000 0,317145 1,404292 0,007802 0,317145
14 0,480740 0,341083 1,510288 0,008390 0,341083
15 0,514337 0,364920 1,615836 0,008977 0,364920
16 0,547781 0,388648 1,720903 0,009561 0,388648
17 0,581062 0,412261 1,825459 0,010141 0,412261
18 0,614170 0,435751 1,929472 0,010719 0,435751
19 0,647096 0,459112 2,032911 0,011294 0,459112
20 0,679829 0,482336 2,135745 0,011865 0,482336
21 0,712359 0,505416 2,237943 0,012433 0,505416
22 0,744678 0,528346 2,339474 0,012997 0,528346
23 0,776774 0,551118 2,440308 0,013557 0,551118
24 0,808639 0,573726 2,540415 0,014113 0,573726
25 0,840263 0,596163 2,639764 0,014665 0,596163
26 0,871636 0,618422 2,738326 0,015213 0,618422
27 0,902749 0,640497 2,836070 0,015756 0,640497
28 0,933592 0,662380 2,932966 0,016294 0,662380
29 0,964156 0,684065 3,028987 0,016828 0,684065
30 0,994432 0,705546 3,124101 0,017356 0,705546
31 1,024411 0,726815 3,218281 0,017879 0,726815
32 1,054082 0,747867 3,311498 0,018397 0,747867
33 1,083439 0,768695 3,403722 0,018910 0,768695
34 1,112470 0,789293 3,494927 0,019416 0,789293
35 1,141168 0,809654 3,585084 0,019917 0,809654
36 1,169523 0,829772 3,674165 0,020412 0,829772
37 1,197528 0,849641 3,762144 0,020901 0,849641
38 1,225172 0,869255 3,848992 0,021383 0,869255
39 1,252449 0,888607 3,934683 0,021859 0,888607
40 1,279348 0,907692 4,019191 0,022329 0,907692
41 1,305863 0,926505 4,102490 0,022792 0,926505
42 1,331985 0,945038 4,184553 0,023248 0,945038
43 1,357705 0,963286 4,265356 0,023696 0,963286
44 1,383016 0,981244 4,344872 0,024138 0,981244
45 1,407909 0,998906 4,423078 0,024573 0,998906
46 1,432378 1,016266 4,499948 0,025000 1,016266
47 1,456414 1,033320 4,575459 0,025419 1,033320
48 1,480010 1,050061 4,649587 0,025831 1,050061
49 1,503158 1,066484 4,722309 0,026235 1,066484
50 1,525851 1,082585 4,793602 0,026631 1,082585
51 1,548082 1,098358 4,863444 0,027019 1,098358
52 1,569844 1,113798 4,931811 0,027399 1,113798
53 1,591130 1,128901 4,998684 0,027770 1,128901
54 1,611934 1,143661 5,064040 0,028134 1,143661
55 1,632248 1,158073 5,127858 0,028488 1,158073
56 1,652066 1,172134 5,190119 0,028834 1,172134
57 1,671383 1,185839 5,250803 0,029171 1,185839
58 1,690191 1,199183 5,309890 0,029499 1,199183
59 1,708484 1,212163 5,367362 0,029819 1,212163
60 1,726258 1,224773 5,423200 0,030129 1,224773
61 1,743506 1,237010 5,477385 0,030430 1,237010
62 1,760222 1,248871 5,529902 0,030722 1,248871
63 1,776402 1,260350 5,580732 0,031004 1,260350
64 1,792040 1,271445 5,629860 0,031277 1,271445
65 1,807131 1,282152 5,677269 0,031540 1,282152
66 1,821670 1,292467 5,722944 0,031794 1,292467
67 1,835652 1,302388 5,766871 0,032038 1,302388
68 1,849073 1,311910 5,809036 0,032272 1,311910
69 1,861929 1,321031 5,849424 0,032497 1,321031
70 1,874216 1,329748 5,888022 0,032711 1,329748
71 1,885928 1,338058 5,924818 0,032916 1,338058
72 1,897063 1,345959 5,959801 0,033110 1,345959
73 1,907618 1,353447 5,992958 0,033294 1,353447
74 1,917587 1,360520 6,024278 0,033468 1,360520
75 1,926969 1,367177 6,053752 0,033632 1,367177
76 1,935760 1,373414 6,081371 0,033785 1,373414
77 1,943958 1,379230 6,107123 0,033928 1,379230
78 1,951559 1,384623 6,131003 0,034061 1,384623
79 1,958561 1,389591 6,153001 0,034183 1,389591
80 1,964962 1,394133 6,173110 0,034295 1,394133
81 1,970760 1,398246 6,191324 0,034396 1,398246
82 1,975952 1,401930 6,207637 0,034487 1,401930
83 1,980538 1,405184 6,222044 0,034567 1,405184
84 1,984515 1,408006 6,234539 0,034636 1,408006
85 1,987883 1,410395 6,245119 0,034695 1,410395
86 1,990640 1,412351 6,253780 0,034743 1,412351
87 1,992785 1,413873 6,260519 0,034781 1,413873
88 1,994318 1,414960 6,265335 0,034807 1,414960
89 1,995238 1,415613 6,268224 0,034823 1,415613
90 1,995544 1,415831 6,269188 0,034829 1,415831
           
  Width 4,427922   Width (180 * Baseline 4,427922
  Height 2,831661   Height 2,831661
  Ratio 1,563719   Ratio 1,563719

From this table the following conclusion can be made:
Using the formula for equal-area cylindrical projection of an ellipsoid with the reference ellipsoid of Friedrich Wilhelm Bessel 1841 and the standard parallel of 45,288534° N/S the result is the same as if Arno Peters method of construction had been used. Both maps would be identical.

 

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Date of last amendment: 08. Februar 2003