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The question, that poses itself, is: Can a map be produced exclusively through a projection? The answer is: No! Arno Peters has developed a system, with which a map can be produced by purely mathematical calculations. Therefore, the type of map-production of Arno Peters is not called projection but "construction". In order to understand the Peters-construction, one must oust the thoughts of any type of projection of the surface of the world and must turn to the mathematical construction completely.
In the following the system is described how the global map is produced using the Peters-construction method.
Definition of terms | ||
![]() Spheroidzone | ![]() Zonestripe | ![]() Zonesegment |
A part of the surface of an spheroid between the equator and any circle of latitude. | A part of the surface of an spheroid between two predefined circles of latitude. | A specific part of the Zonestripe |
For demonstration the circles of latitude are drawn in steps of 10 ° and the circles of longitude are drawn in steps of 20°. |
A grid mesh is a rectangular area from which a grid is constructed.
Starting point for our global map is a horizontal line with a length of the desired width of our global map. Let's consider our map should be 3,60 m (= 360 cm) wide. This line represents the equator. Because there are twice a many circles of longitude than circles of latitude a zonesegment is twice as wide than high. Every zonesegment is l = 2° wide and j = 1° high.
The width of our basic grid mesh is 360 cm / 180 = 2 cm wide. Because the basic grid mesh is a square the height is also 2 cm.
First we determine the geocentrical latitude of the geographical latitude j = 1° and the result for y = 0,993306966°. Then we put all values in to the formula and determine the area of the spheroidzone = 4431047,0014894 km².
Now we divide this area by 180 and the result is the area of the zonesegment 4431047,001489400 / 180 = 24616,927786052 km².
To get a square we draw the square root of this area and we have the basic grid mesh height Ö24616,927786052 km² = 156,897825944 km. This height is called baseline and will be used to determine all heights of all following grid meshes.
On our global map this value equals the ratio 2 cm / 156,897825944 km = 0,012747149.
Because of the conversion to a square a distortion in form is the result.
Estuary of the Amazonas j = 0°N to 1°N, l = 51°W to 49°W | |
Zonensegment | ![]() Basic grid mesh |
All grid meshes within the grid have the same width (baseline).
All grid meshes between two circles of latitude have the same height.
To get the height of the next grid mesh we first have to determine for j = 2° equals y = 1,986622004°. The area of the spheroidzone is 8860780,38317385 km². To get the area of the zonestripe between j = 1° to 2° we have to subtract the area of the preceding spheroidzone from the actual spheroidzone 8860780,38317385 km² - 4431047,0014894 km² = 4429733,38168445 km². The area of the zonesegment is 4429733,38168445 / 180 = 24609,629898247 km²: The height of this zonesegment equals the ration between the area of the zonesegment and the baseline 24609,629898247 km² / 156,897825944 km = 156,851312312 km.
On our map this value equals 156,851312312 * 0,012747149 = 1,999407084 cm.
The height of a grid mesh equals the ratio between the area of the zonesegment and the baseline.
All following grid meshes are calculated with the same scheme.
Earth's surface | Our global map | ||||||
j | y | Spheroidzone | Zonestripe | Zonesegment | Height | Height | Cumulated Height |
(°) | (°) | Z (km²) | Zx - Zx-1 (km²) | A = Zx / 180 (km²) | h (km) | h (cm) | hcum (cm) |
1 | 0,993307 | 4431047,001489 | 4431047,001489 | 24616,927786 | 156,897826 | 2,000000 | 2,000000 |
... | |||||||
10 | 9,934394 | 44093962,579799 | 4372034,074873 | 24289,078194 | 154,808252 | 1,973364 | 19,902277 |
20 | 19,876630 | 86881830,441838 | 4183281,499747 | 23240,452776 | 148,124760 | 1,888168 | 39,215034 |
30 | 29,833636 | 127088269,973908 | 3869280,153550 | 21496,000853 | 137,006365 | 1,746441 | 57,362637 |
40 | 39,810611 | 163500855,294729 | 3437827,233588 | 19099,040187 | 121,729158 | 1,551700 | 73,797843 |
50 | 49,810390 | 195004372,156136 | 2900248,357196 | 16112,490873 | 102,694163 | 1,309058 | 88,017289 |
60 | 59,833076 | 220616960,330178 | 2271530,040678 | 12619,611337 | 80,432034 | 1,025279 | 99,577802 |
70 | 69,875993 | 239526470,178184 | 1570231,881968 | 8723,510455 | 55,599945 | 0,708741 | 108,112809 |
80 | 79,933979 | 251124238,080918 | 818078,072684 | 4544,878182 | 28,967120 | 0,369248 | 113,347585 |
90 | 90,000000 | 255032810,842458 | 39192,005387 | 217,733363 | 1,387740 | 0,017690 | 115,111761 |
This is just an extract of the complete table | |||||||
All 16200 calculated grid meshes (including the 180 basic grid meshes) are now mirrored horizontally around the equator. so we get the grid for the southern hemisphere.
Remark: The grid has to be mirrored because the valid range for the formula to calculate the area of the spheroidzones goes from y = 0° to y = 90° only.
The total height of our global map is htot = 2 * hNorth = 2 * 115,11176061 cm = 230,223521219 cm. The ratio between width and height equals 360 cm / 230,223521219 cm = 1 : 1,563697741.
For the contemplation of the precision, two aspects are imperative. To the one the user of the map, who expects, that the surface of the earth is represented, as shown on the map. A pilot compares the surface of the earth below him with the map and determines his position. Depending on the use of a map the scale of the map will vary. Therefore, the surveyor who surveys a property will uses a map with a larger scale than the climate-researcher, who presents a report on the global warming of the seas.
On the other side are the cartographers, whose task is it to draw geographical data of the world on a two-dimensional map. For the drawing of coordinates predefined lines are required.
Since the width of the Peters global map can be chosen arbitrarily, all longitudes behave to each other linearly. The reference-line for the drawing of longitudinal-coordinates is any circle of longitude. No problem.
The reference-line for the drawing of latitudinal-coordinates is the equator. The height of the grid meshes decreases toward the poles. Latitudinal-coordinates do not behave linearly within even one grid mesh.
In order to draw any latitudinal-coordinate, the ration of the area of the spheroidzone, from which the basic grid mesh was developed, to the baseline must be known first. Area of the spheroidzone / baseline = 4431047,0014894 km² / 156,897825944 km = 28241,608669977 km.
If for example the geographical latitude 30° needs to be drawn, the geocentrical latitude is calculated first. With this value the area of the spheroidzone is determined. Then, this area 127088269,973908 km² is divided by 28241,608669977 km divides = 4500,036504968 km.
This value corresponds to on our map 4500,036504968 km * 0,012747149 = 57,362635834 cm.
This result can also be achieved by adding the heights of all 30 grid meshes. However problems will occur using this method when decimal numbers need to be drawn (e.g. 37,839° N). For any 1/10 degree precision the number of grid meshes that need to be calculated will increase by 10.
Simultaneously with the introduction of the mathematical principles of constructing his global map Arno Peters established a decimal grid. He relocated the reference longitude 0° to a position between the continents Russia and America arbitrarily. His new longitude 0° runs almost exclusively over water. From there he counts 100 circle of longitude to the right (east). The reference line for the circle of latitudes is the north pole. From there he counts 100 circles of latitude downwards to the south pole. For expressing any geographical coordinate the terms northern and southern latitude or easterly and westerly longitude is not required anymore.
![]() On the Peters-map the Longitude 0 runs through the Behring-Sea. |
The conventional longitude 0° runs through the very easterly part of Russia. |
New York has the geographical coordinates » 41° northerly latitude 74° westerly longitude. |
If one intended to establish the decimal grid for the use on maps all geographical data would need to be recalculated worldwide. A hopeless venture.
The conventional 360° grid is based on a long historical development by the observation of the sun and the stars. This has influenced our calendar (365 days for one orbit around sun), as well as also our understanding of the time (24 hours for one rotation around the poleaxis). Should the decimal grid be established our calendar and the definition of time has to be adapted inevitably. In present days this seems to be impossible to me.
In his atlas Arno Peters used the conventional 360° grid. Only in the rear book-cover his global map with the decimal grid is found. Also on his global map "All countries of the world projected with fidelity of area" the conventional grid is used. Only on the maps edges, the division according to his decimal system can be found.
Source http://www.heliheyn.de/Maps/HowDo_E.html