Miles and Pixels and Mollweide Projections

[Zirojtan]

Hey guys, I'm wondering how I calculate how many miles a pixel on my map represents when the map is a Mollweide Projection when the actual image of the planet itself (minus the black background) is 4970 pixels in width (at the planet's widest point) and 2515 pixels in height?


How would I figure this out mathematically?


Thanks

[waldronate]

The width of a Mollweide projection is the equatorial circumference of your planet; the height is the distance from north pole to south pole (half the equatorial distance if your planet is spherical). If we assume that your planet is 40000km in circumference (you didn't specify and we'll go with Earth-sized), then the number of km/pixel horizontally is 40000/4970 and the number of km/pixel is 20000/2515 pixels vertically (or about 8 km/pixel horizontally and vertically).

[Azélor]

Logically you might think of this

At the equator :
4970 = 40 000 km
40 000/4970=8 km

From pole to pole:
2515 = 20 000 km
= 8 km (more or less)

With each pixel representing 8 km by 8 km or 64 square kilometres
But it's wrong

If your planet is as big as Earth, it covers an area of more or less 510 000 000 square kilometres.
The problem is that if you multiply the area covered by 1 pixel (64 km2) with the number of pixels in the ellipse, you will have an area larger than Earth by about 22%.
So to find out the real size you have to divide the total area of the world by the number of pixels in the ellipse.

the result should be close to 64/1,22
= 52,46 km2 or 20,255 sq miles for each pixel

[Zirojtan]

Thank you

[Hai-Etlik]

Do not think of pixels as having meaningful side lengths. They have a fixed area but significantly varying shapes, this is why the naive calculation Azelor warned against is off by 22% as the pixels near the equator are not square. You should be thinking in terms of areas only, never distances on such a map. It should not have any form of linear scale indicator like a scale bar. If that's what you were planning, you are out of luck. You can include an area scale though. This would be a simple shape like a circle and an indication of the area it represents. You might include several if you wanted to.

For the same reason, you shouldn't include a compass rose or rhumb lines as Mollweide does not preserve bearings. I'm sure someone will join the thread to tell me I'm a horrible excuse for a human being for pointing this out.

To find the area per pixel, simply divide the total area but the total number of pixels as Azelor said.

[waldronate]

Silly me, I interpreted "miles per pixel" as a linear distance (and then I gave the distance in km/pixel for some reason...)

As a wise woman once said "Math class is HARD."