3 ms·

*Spherical cap https://en.wikipedia.org/wiki/Spherical_cap https://en.wikipedia.org/wiki/Spherical_cap Bonus points for whoever calculates the ratio of the su

*Spherical cap https://en.wikipedia.org/wiki/Spherical_cap https://en.wikipedia.org/wiki/Spherical_cap Bonus points for whoever calculates the ratio of the surface areas for the two spherical caps that contain half of the human population. (The circle's radius is ~4,000 km)

Waterluvian 7y ago
Can I cheat and use QGIS? Feel like if I got up off my butt and went to my PC it would take two mins. I assume we are simplifying to an oblate spheroid and not worrying about topography and such. Also I think this means technically its a Spheroidal Cap?
- symplee 7y ago
  If we bring in topography, aren't we now running into a 3D version of the coastline paradox? https://en.wikipedia.org/wiki/Coastline_paradox https://en.wikipedia.org/wiki/Coastline_paradox So, yes, good point. Let's start with assuming a perfect sphere, then oblate spheroid, then topography, then only land (not water), and however crazy this thread gets with whatever else we haven't thought of :)
thaumasiotes 7y ago
> Bonus points for whoever calculates the ratio of the surface areas for the two spherical caps Hmpf, I went to all the trouble of deriving the surface area of a spherical cap (it's really easy, as surface integrals go), and it's listed right there in the wikipedia page. A = 2πrr(1 - cos φ), where φ is the angle between the center of the cap and the edge (measured from the center of the sphere). Assumptions: 1. The "radius" of the cap is 4000 km. 2. The "radius" of the cap is actually an arclength along a great circle of a sphere (the earth). 3. The radius (a real radius) of the earth is somewhere between 6300 and 6400 km. The formula for arclength along a circle tells us that φ is somewhere between 40/63 and 40/64. We can calculate the cap's percentage of the surface area of the entire sphere as (1 - cos φ) / 2. For an earth-radius of 6300km, this is 9.744%. For 6400km, this is 9.452%. For the smaller earth, the ratio of the cap's outside to its inside is 9.26:1; for the larger earth, it is 9.58:1. Rounding to two significant figures, like we used for the size of the earth, the ratio should be between 9.3:1 and 9.6:1. The ratio isn't really very informative, because the cap was drawn around where a bunch of people live, and "everything else" wasn't. This has artifactually put most of the ocean, where people cannot live, into "everything else". "Everything else" should probably be significantly deflated to adjust for this. The article messes up a similar point: > The population density of Greenland, for example, is just 0.1/sq. mi—that is, one person living on every ten square miles of rock and ice. But it's a lot easier to find company in Manila, which is literally one million times as crowded: 107,000 Filipinos per square mile. The population density of Greenland is mostly zero with some spikes. There's not a literal tenth of a person every square mile -- those people would all be dead. The population density in areas where local population density is more than zero is much higher. Manila is not actually one million times as dense.
- thaumasiotes 7y ago
  It's more impressive to compare the area of the cap to the area of a flat circle of the same radius. The curvature really adds up -- the cap is under 40% of the circle.
  - deleted 7y ago
    [deleted]
  - thaumasiotes 7y ago
    Ah, the reason this was so impressive is that I mixed up the radius of the cap with the radius of the sphere on which it sits. In reality, this cap has 97% of the area it would if it were a flat circle of the same radius.

Waterluvian 7y ago

Can I cheat and use QGIS? Feel like if I got up off my butt and went to my PC it would take two mins. I assume we are simplifying to an oblate spheroid and not worrying about topography and such. Also I think this means technically its a Spheroidal Cap?

symplee 7y ago
If we bring in topography, aren't we now running into a 3D version of the coastline paradox? https://en.wikipedia.org/wiki/Coastline_paradox https://en.wikipedia.org/wiki/Coastline_paradox So, yes, good point. Let's start with assuming a perfect sphere, then oblate spheroid, then topography, then only land (not water), and however crazy this thread gets with whatever else we haven't thought of :)

thaumasiotes 7y ago

> Bonus points for whoever calculates the ratio of the surface areas for the two spherical caps Hmpf, I went to all the trouble of deriving the surface area of a spherical cap (it's really easy, as surface integrals go), and it's listed right there in the wikipedia page. A = 2πrr(1 - cos φ), where φ is the angle between the center of the cap and the edge (measured from the center of the sphere). Assumptions: 1. The "radius" of the cap is 4000 km. 2. The "radius" of the cap is actually an arclength along a great circle of a sphere (the earth). 3. The radius (a real radius) of the earth is somewhere between 6300 and 6400 km. The formula for arclength along a circle tells us that φ is somewhere between 40/63 and 40/64. We can calculate the cap's percentage of the surface area of the entire sphere as (1 - cos φ) / 2. For an earth-radius of 6300km, this is 9.744%. For 6400km, this is 9.452%. For the smaller earth, the ratio of the cap's outside to its inside is 9.26:1; for the larger earth, it is 9.58:1. Rounding to two significant figures, like we used for the size of the earth, the ratio should be between 9.3:1 and 9.6:1. The ratio isn't really very informative, because the cap was drawn around where a bunch of people live, and "everything else" wasn't. This has artifactually put most of the ocean, where people cannot live, into "everything else". "Everything else" should probably be significantly deflated to adjust for this. The article messes up a similar point: > The population density of Greenland, for example, is just 0.1/sq. mi—that is, one person living on every ten square miles of rock and ice. But it's a lot easier to find company in Manila, which is literally one million times as crowded: 107,000 Filipinos per square mile. The population density of Greenland is mostly zero with some spikes. There's not a literal tenth of a person every square mile -- those people would all be dead. The population density in areas where local population density is more than zero is much higher. Manila is not actually one million times as dense.

thaumasiotes 7y ago
It's more impressive to compare the area of the cap to the area of a flat circle of the same radius. The curvature really adds up -- the cap is under 40% of the circle.
- deleted 7y ago
  [deleted]
- thaumasiotes 7y ago
  Ah, the reason this was so impressive is that I mixed up the radius of the cap with the radius of the sphere on which it sits. In reality, this cap has 97% of the area it would if it were a flat circle of the same radius.