The making of Land surface

About the spherical Mercator math

We have a map of 512×512 tiles where each pixel has a unique hex code. Dividing the map into 512 lines, each with 512 tiles. The line is 131 072×256 pixels in size, 33 554 432 pixels overall.

All that’s left is to calculate how many square kilometers are made up by pixels in those lines.

First, understanding where the Mercator projection put our lines on the sphere. Learning that on Google Maps data is shown in a spherical coordinate system (Spherical Mercator EPSG:900913 (EPSG:3857) and WGS84 Datum). The sphere radius is 6 378 137 meters. The map does not show the entire sphere, the maximum polar parallel is 85,051 128 78°.

In the Mercator projection, the parallels are shifted the further they are from the central meridian.

This parallel shift (y) is represented by formula 1:

R=1; θ is the parallel angle in radians.

The resultant parallel shift value (scale coefficient) is converted back into radians using formula 2:

On a Mercator sphere the scale of map segments (tiles) is decreased gradually with distance from equator.

If we were to imagine the Mercator projection as a gradient scale, it would be absolutely linear.

All lines on the scale are exactly the same, yet the closer the line to a pole on the map, the smaller its scale on the sphere. To determine the scale range, we convert the angle of the maximum polar parallel into radians 85,051 128 78° · π / 180° = 1,484 422 229 748 71 and calculate its scale coefficient using formula 1.

1 · ln (tan(1,484 422 229 748 71 / 2 + π / 4)) = 3,141 592 653 628 9

Since the sphere is symmetrical, it is sufficient to determine the scale coefficient for each parallel for only half of the sphere, meaning to calculate angles for 256 lines that have the same size on the map and become smaller the further they are from the equator on the sphere. The scale gradient can be divided into any number of segments, but given the scale of the map we need exactly 256, from the equator (y₂₅₅) to the polar parallel (y₀).

Dividing the scale gradient into 256 segments.

y₂₅₅ = y₀ / 256 = 0,012 271 846 3
y₂₅₄ = y₂₅₅ + y₂₅₅ = 0,024 543 692 6
y₂₅₃ = y₂₅₄ + y₂₅₅ = 0,036 815 538 9
y₂₅₂ = y₂₅₃ + y₂₅₅ = ...
y₂₅₁ = y₂₅₂ + y₂₅₅
And so on all the way to the polar parallel.

Using formula 2, we can convert the scale coefficient for each parallel into radians and degrees.

θ₂₅₅ = 2 atan (exp (y₂₅₅ / 1)) – ½ π = 0,012 271 538 3 rad = 0,703 107 352 4°
θ₂₅₄ = 2 atan (exp (y₂₅₄ / 1)) – ½ π = 0,024 541 228 8 rad = 1,406 108 835 5°
θ₂₅₃ = 0,036 807 225 2 rad = 2,108 898 659 3°
etc.

Calculating the area of the polar segment that didn’t fit on the map.

Polar angle (φ₀) = (90° – θ₀) · 2 = 9,897 742 44° = 0,172 748 194 1 rad
Polar segment height = Sphere radius · (1 – cos(φ₀ / 2)) = 23,777 208 59 km
Polar chord = 2 · Sphere radius · sin(φ₀ / 2) = 1 100,442 151 23 km
Polar area = π · ((Polar chord / 2) ^ 2 + Polar segment height ^ 2) = 952 872,030 890 85 km²

All that’s left is to calculate the area of each line.

Finding the angle (φ) of intersection of the parallel with the sphere radius in radians for each parallel from θ₂₅₅ to θ₀.

φ₂₅₅ = (90° – θ₂₅₅) · 2 π / 180° = 3,117 049 577 rad
φ₂₅₄ = (90° – θ₂₅₄) · 2 π / 180° = 3,092 510 195 9 rad
φ₂₅₃ = 3,067 978 203 2 rad
etc.

We already know the polar segment height, now we need to calculate the height (h) of the lines:

h₀ = Sphere radius · (1 – cos(φ₀ / 2)) – Polar segment height = 0,589 672 519 79 km
h₁ = Sphere radius · (1 – cos(φ₁ / 2)) – Sum of previous segments' heights = 0,604 267 661 96 km
h₂ = Sphere radius · (1 – cos(φ₂ / 2)) – Sum of previous segments' heights = 0,619 222 612 38 km
h₃ = 0,634 546 168 98 km
etc.
h₂₅₅ = Sphere radius – Sum of previous segments' heights = 78,267 588 018 18 km

Finally, we get the area (S) of each line.

S₀ = 2 π · Sphere radius · h₀ = 23 631,136 069 7 km²
S₁ = 2 π · Sphere radius · h₁ = 24 216,036 635 6 km²
S₂ = 24 815,356 523 4 km²
S₃ = 25 429,448 309 54 km²
etc.

S₂₅₅ = 3 136 574,895 768 57 km²
S₂₅₆ = S₂₅₅
S₂₅₇ = S₂₅₄
S₂₅₈ = S₂₅₃
etc.
S₅₁₁ = S₀

To get the area of a pixel in the line, we simply divide the line area by the number of pixels it has.

Tiles in the Mercator projection in scale 10 on the sphere have a shape that looks oddly like a square.

Using some magic to calculate the number of pixels.

Looking at how many square kilometers are made by pixels in each line.

Rounding the numbers and comparing with the data from Wikipedia.

Recalling our story about Yemen and the housewife.

Some territories didn’t make it on our list, including the Somali Republic, Jubaland, Galmudug, Puntland, Maakhir, Khatumo, Somaliland, Awdalland, Nagorno-Karabakh Republic, Lugansk and Donetsk People’s Republics.

Part IV