We have all seen those signposts - the ones stuck in the middle of nowhere with a multitude of signs pointing to famous places (London, New York, Tokyo, Rowville, etc).
They all point across the surface of the earth - presumably to follow a great circle to the desired locale.
I want one which points in a true straight line. As the laser shoots. THROUGH THE EARTH! (or is that "as the mole people dig!"?)
Make a hole with the gun perpendicular / To the name of this town in a desktop globe
Exit wounds in a foreign nation / Showing the home of the one this was written for
—They Might Be Giants (Ana NG)
Especially from Australia, this would be pretty neat I think, with North America and Europe being the most 'down' you can get from an Australian perspective. More interesting stuff would be in the directions of Africa, Asia, south America, Antarctica, NZ...
Tell me more, tell me more
For bonus points, instead of a signpost, make it out of actual lasers! (and sealed in a cage with appropriate gas so the line of the laser is visible. Say, ALL lasers always visible in red, then you select a city and it goes GREEN!).
Another method for a permanent exhibit might be to start with a point central to the room, (normal signpost), but then ALSO place markers on the walls and floor of the room to indicate the marker - so as to make clearer the path and descent taken.
Bearing and Arc Distance
The forumulas to calculate from raw data are given in links below, but for example, this site (http://gc.kls2.com/) will calculate the bearing and great circle distance. The required final data then requires a conversion: the great circle distance into both a straight line (distance), and angle down (0 being tangent to here, 90degrees being straight down). The degrees should remain the same.
Declination and Direct distance
So knowing the arc distance, we can calculate the declination (angle down from tangent to source location), and direct distance via the following:
- We assume the earth is a sphere, with radius 6371km (the avg radius of the earth, whcih varies in fact between 6356.8 and 6378.1km)
- The declination (angle down) is calculatable from knowing the center angle and distances of the isosceles trangle with points of the two locations, and the center of the earth. In this instance, we know the arc length between two locations, so the center angle can be found with: θ = s/r, where θ is the subtended angle in radians, s is arc length, and r is radius
- From the internal angle, the outer angles are discoverable (π radians in a triangle, thus (π-θ)/2 is the internal angle "up" from drop-down angle. We take that from the tangent angle and we get:
- In degrees: One radian is 180/π, so the inner angle θ=s/r becomes a=180s/πr
- Because we know r=6371km, we can solve to: a=s*0.00899321606...
- The other inner angle is then b=(180-a)/2, so our final formula becomes: 90-b
- With everything plugged in: 90-((180-(s*0.00899321606))/2)
In shell script:
echo "scale=2;90-((180-($arckm*0.00899321606))/2)" | bc
- Not dissimilarly, knowing two sides (r) and all three angles (a,b,b), we can calculate the third edge length (d) (using the rule of sine)
The raw data from Canberra (CBR) is:
(airport or Lat/Lon)
|Bearing||Arc length||Declination||Direct distance|
|North Pole||0E,90N||0° (N)||13911km||62.56°|
|Los Angeles||LAX||62° (NE)||12282km||55.23°|
|New York City||JFK||67° (E)||16239km||73.03°|
|Mexico City||MEX||85° (E)||13178km||59.26°|
|Miami||MIA||88° (E)||15213 km||68.41°|
|Galapagos Islands||GPS||109° (E)||12710km||57.16°|
|Easter Island||IPC||118° (SE)||9255km||41.62°|
|São Paulo||SBGR||163° (S)||13298km||59.8°|
|South Pole||NZSP||180° (S)||6093km||27.4°|
|Cape Town||CPT||219° (SW)||10780km||48.48°|
| Las Palmas
[Furthest land from CBR?]
|New Delhi||DEL||302° (NW)||10350km||46.54°|
- Make a script which can calculate this between any two points, given their long and lat.
- Make a physical sign for same
- Make a smartphone app for the same too (assuming it has a GPS, a compass, and a 3axis accelerometer. eg, iPhone 3GS)