GPS coordinates: 1km square around a point

Question

I was hoping someone out there could provide me with an equation to calculate a 1km square (X from a.aaa to b.bbb, Y from c.ccc to c.ccc) around a given point, say lat = 53.38292839 and lon = -6.1843984? I'll also need 2km, 5km and 10km squares around a point.

I've tried googling around to no avail... It's late at night and was hoping someone might have quick fix handy before I delve into the trigonometry...

I'll be running all this in Javascript, although any language is fine.

Solution 1

If the world were a perfect sphere, according to basic trigonometry...

Degrees of latitude have the same linear distance anywhere in the world, because all lines of latitude are the same size. So 1 degree of latitude is equal to 1/360th of the circumference of the Earth, which is 1/360th of 40,075 km.

The length of a lines of longitude depends on the latitude. The line of longitude at latitude l will be cos(l)*40,075 km. One degree of longitude will be 1/360th of that.

So you can work backwards from that. Assuming you want something very close to one square kilometre, you'll want 1 * (360/40075) = 0.008983 degrees of latitude.

At your example latitude of 53.38292839, the line of longitude will be cos(53.38292839)*40075 = [approx] 23903.297 km long. So 1 km is 1 * (360/23903.297) = 0.015060 degrees.

In reality the Earth isn't a perfect sphere, it's fatter at the equator. And the above gives a really good answer for most of the useful area of the world, but is prone to go a little odd near the poles (where rectangles in long/lat stop looking anything like rectangles on the globe). If you were on the equator, for example, the hypothetical line of longitude is 0 km long. So how you'd deal with a need to count degrees on that will depend on why you want the numbers.

Solution 2

Here is something from my notes to be used on Android with its decimal GPS.

Lat Long: NY City 40N 47 73W 58 40.783333 73.966667

Wash DC 38N 53 77W 02 38.883333 77.033333

yields = 209 miles !! VERY CLOSE

Distance (miles) (x) = 69.1 (lat2-lat1) Distance(miles) (y) = 53.0 (long2 - long1) As crow flys sqrt (x2 + y2) ... duh!@

delta(LAT) / Mile = .014472 delta(LONG) / Mile = .018519

Using a box as approximation To find someone within 100 miles (100 north / 100 south, 100 E / 100 W) From 0,0 -14.472 / + 14.472 , -18.519 / 18.519

Solution 3

A simpler way of generating a gps square given the centre would be to use the indirect Vincenty algorithm.The Javascript code here shows how to do it http://www.movable-type.co.uk/scripts/latlong.html. Creating a square using a circle isn't to hard. Squares are equal distance to each point. So given a centre point, distance from the centre, change the bearing from 0 or any number depending on rotation of the square and increment by 90 degrees or PI/2 radians. By incrementing by 90 degrees each time and you will up with a square in circular space.

I use this myself for generating GPS points around a centre point with a given distance .---. --/- --0-- -/-- .---.