anti neighbours (pdf)
TRANSCRIPT
Not long ago I had the privilege of going to Germany, which is just about as far as it is possible to get from my home in New Zealand. It is not the absolute furthest distance from my home but short of it by just a few percent.
To get to the furthest country from New Zealand I would have to travel a bit to the West and South of Germany, to the north of Spain.
It’s pretty easy to calculate where your anti-neighbour lives: just add 180 degrees to your home’s longitude and reverse the sign of your home’s latitude. So for example if you live at longitude 10 degrees then your anti-neighbour lives at longitude 190 degrees. If you live 30 degrees south of the equator then your anti-neighbour lives 30 degrees north of the equator.
You can find your own latitude and longitude easily enough by typing that question into your laptop’s search engine.
But to do the job really properly you need to have the latitude and longitude of the house you are living in, not just an average point for the city you live in.
You can probably find your exact latitude and longitude in a variety of ways these days with GPS technology but I found it in a matter of minutes using Google Maps. Here for example are the coordinates for the Grand Canyon.
Type in your home address and you will find a curious string of numbers appearing in the location box at the top of the screen. This is your exact latitude and longitude separated by a comma and shown to an accuracy of seven decimal places.
So carefully cut and paste these numbers somewhere and make the adjustments I mentioned earlier and you will have the exact geographic coordinates of your anti-neighbour. ....
... Then feed this back into Google Maps (as an address) and you will be taken to a point in space directly above your anti-neighbour’s house. Zoom in and switch to satellite view and you will actually see his/her house, the house of the person who is farthest away from you in the entire world.
I’ve done this for my own house and found that my anti-neighbour lives in a farm house, the nearest building to the point in the middle of a field which is directly opposite my house in New Zealand.
It would be very tempting to say that this person is as far away from me as it is possible to get but that is only true if we are thinking of the journey across the surface of the world. If we think in terms of the linear distance through the middle, that is a much more complex matter.
The world is not perfectly round. It bulges in the middle (meaning along the equator) as a result of the turning of the Earth. That means that people who live near the equator are a little further away from everybody else than they ought to be if we treat the Earth as a sphere, and people who live near the poles are a little bit closer.
The person who lives furthest from us (by straight line through the interior of the Earth) is therefore a little bit closer to the equator than (s)he ought to be if the world were considered a sphere.
In my case, this turns out to be a shift of 100 or so km. My spherical anti-neighbour is about a dozen km south of the north coast of Spain but my straight-line anti-neighbour is somewhere in Portugal, near its inland border.
To work this out I had to slice the world through the middle and treat the exposed flat surface as an ellipse.
Knowing my own latitude I could plot my position on the surface of that ellipse and then draw a line through the interior to the other side of the ellipse. Then it becomes a matter of swinging that line back and forth and seeing how the length of the line changes. This is best done on a spreadsheet of course, using the mathematical definition of an ellipse and Pythagoras's rule to calculate the distance in any given direction.
Home
To get the distance through the interior of the Earth to any desired point on the surface requires a bit of spherical trigonometry, which is not as hard as it sounds even though the formulae at first glance look scary.
But before I introduce that procedure, let me show you a quick and easy thing I did to work out the straight-line distance through the interior of the Earth to my apartment in Darmstadt from my home in NZ.
Home North of Spain
Darmstadt
Google Maps comes to the fore once again. Knowing where my anti-neighbour lives in the north of Spain, I can ask GM to plot the best road journey from there to Darmstadt and tell me the distance.
That of course is a twisty journey but I can eyeball the straight line distance from the mapped road journey and estimate what that would be in comparison. If I was really keen I could get a ruler out and hold it up to the screen but that is hardly necessary since a rough estimate will do.
Let’s say the distance turns out to be about 2000 km (it’s not but that number is easy to fling around as I explain this). Divide that by the circumference of the Earth, which is about 40,000 km and you will have the fraction of a circle that separates Darmstadt from my Spanish anti-neighbour. Multiply this by 360 and you will have the angular separation of these two points. That’s about 18 degrees.
Home North of Spain
Darmstadt
18O
Home North of Spain
Darmstadt
18O
Now the clever bit: The angle from under my toes in NZ to the foundations of my apartment in Germany is half of the angle made at the centre of the Earth. This is a property of circles which is easily proven and is very convenient because now I can use ordinary trigonometry to compare the distance to Germany with the distance to my anti-neighbour in Spain. Simply put, it’s the cosine of the half angle times the diameter of the Earth.
9O
In this case the distance through the Earth to Germany is shorter than the distance to Spain by a few hundred kilometres. This will be affected somewhat by the non-sphericity of the Earth but the distortion will affect both distances to a similar extent so that the difference between those numbers will not be very big.
Distance = (2 x 6378 km) x Cos( 9O ) = 12599 km
Difference from Earth’s diameter ~ 150 km
It’s important to be aware that this number can’t be made more accurate without bringing a lot more data into consideration. So don’t get caught up with the decimal places in the answer: they’re swamped in error.
If the angle that separates your target from your anti-neighbour’s home is very small (meaning less than about fifteen degrees) then you can take advantage of some small-angle approximations and produce a nice little formula that gives you the distance short of a complete diameter.
2
21 1 ~ Cos
diametershortdistradius
separationsurface x x ~
2
21
21
s x radiu
separation surface ~
4
2
25512
2S D ~ (all measurements in km)
That is a matter of interest to me because my anti-neighbour is not exactly at the point on the globe opposite to me. Rather, (s)he lives in the nearest house to the paddock which is directly opposite my home. That paddock is about a hundred metres from my anti-neighbour. So using that distance to work out the fraction of the circumference not travelled and then using the cosine of the half angle blah blah blah, I can work out how many millimetres closer my anti-neighbour is to me when (s)he is standing in his/her living room compared to standing in that field. It is comparable to the thickness of a blade of grass.
That comparison to the thickness of a blade of grass has implications. If my anti-neighbour mows his lawn then he is now closer to me than the neighbour who hasn’t mowed his lawn. He might regain that title if he puts his boots on and thereby adds a few millimetres to the distance, but if someone a few kilometres away decides to get up and paint his roof then HE is now more distant from me than any of the people I have discussed.
I can use the short-distance formula to determine the size of the circle in which the furthest distance through the Earth shrinks by as much as 10 m. In this circle all residents take turns being my anti-neighbours, whether they live in a two-storey flat or in a basement.
25512
2S D ~ DS ~ 25512
01.025512
km 16
This all falls into irrelevance if the topography of the landscape is uneven. Whoever lives at the top of any mountain in the region is further from me than anyone living lower down. Beyond a certain level of accuracy the analysis becomes meaningless. All I can really say is that the person who lives furthest from me in the world probably lives at the top of a mountain somewhere in Spain or in Portugal.
To get the linear distance (as the worm digs) from any point on the Earth’s surface to any other point on the Earth’s surface, draw a triangle on the surface of the Earth made up of these three sides: () The direct flight path from your home to your destination (BC). () The line extending from your home to the north pole (CA). () The line extending from your destination to the north pole (BA).
These ‘sides’ are all angles. The flight path angle is not known but it can be calculated from the two angles that go to the north pole. These are obtained (in one case) by adding 90 degrees to the latitude and (in the other case) subtracting the latitude from 90 degrees. Another angle is needed to complete the formula, and that is the difference in longitudes of the two points on the surface. You can get those numbers by looking at a world map or searching the internet.
The formula gives you the angle under the flight path, which you can use to get the straight line distance through the Earth. Simple geometry shows that the straight line distance is actually the base of a triangle with two equal sides. The sine of half the angle (times the radius of the Earth) gives you half the distance. Of course double that and the distance is found.
r Sin 2
2
rr
So for example the straight-line distance between Christchurch NZ and Singapore would be found as follows:
Data for Singapore: Latitude = 1O N Longitude = 104O E
Data for Singapore: Latitude = 1O N Longitude = 104O E
On this diagram (that I obtained from Wikipedia – thank you) Singapore would be at point B and Christchurch would be at point C. The north pole would be at point A.
The latitude separation of Singapore from the north pole is 89O. (This is arc c) The latitude separation of Christchurch from the north pole is 224O. (This is arc b)
The longitude separation between Christchurch and Singapore is 69O. Be careful now! This is not arc a. This is angle A. Arc a corresponds to the shortest flight path taken by an aircraft between these two places. That’s what I want to work out from the other information.
ACoscSinbSincCosbCosaCos
OOOOO 96 242 98 242 98 CosSinSinCosCosaCos
0.3584 0.6947- 0.9998 0.7193- .017450 aCos
O105 a This is the angular separation of Christchurch and Singapore.
As a by-product I can get the distance across the Earth’s surface:
km
km dist
11666
40000 360105
km Sin km strline 10120637822
105
105O
strline
So a magical tunnel to Singapore would be 1546 km shorter than a journey over the surface. That’s a decrease of 13 %.
km 11666
Knowing the angle of arc between Singapore and Christchurch, I can work out how much longer (in terms of kilometres) the journey is by air than by sea. In one case the circle radius is about 6378 km and in the other case the radius is about 10 km greater.
kmrs O 11682 6378 105 18011
kmrs O 11701 6388 105 18022
So flying at cruise altitude actually makes the journey 19 km longer.
In principle, this mathematics also describes the journey from my kitchen to my living room. If the floor is horizontal then it is not flat but a segment of the circle that defines the Earth's surface. If I walk from one room to the other, the top of my head has travelled slightly further than my feet. I wonder how much that extra distance would be? It can be calculated even if it is too small to measure.
[END]