Post By Hai-Etlik
Large Scale Graticules Made Easy
First, for those not familiar with cartographic jargon, a graticule is a grid indicating a coordinate system on a map -- normally latitude and longitude. Large scale means "zoomed in". You can think of this two ways, the scale is the ratio between size on the map and size in real life, and 1/2000 is a larger number than 1/100000. Or you can think of the features on the map being drawn larger than at smaller scales.
For the purposes of this tutorial, large scale is where the area covered by the map (the extent) is small enough that the curvature can be ignored; in the vicinity of 600-700km squared is a good upper limit.
So, to start, you should have a map. You should know its scale and extent, you should know where it is on the planet (the latitude and longitude), and you should know how big your planet is.
Now, the distance between lines of latitude ("Parallels") is determined by the size of your planet. So take the radius, multiply by pi, and divide by 180 (πr/180) to get the distance corresponding to 1 degree of latitude.
Next, to get the distance between lines of longitude ("Meridians") take the latitude of the map, and find its cosine. A moderately capable calculator should have a "cos" button to find this. Multiply this by the previous distance to get the distance between meridians.
For small scale maps, spacings of 5, 10, 15, or even 30 or 45 degrees are common. But for a large scale map, 1 degree is probably as coarse as you will want to get. To go smaller, use minutes and seconds; there are 60 minutes in a degree, and 60 seconds in a minute. Again, divisions of 5, 10, 15, or 30 are common, whatever makes for a well spaced grid on your map.
If all that seems rather abstract, here's an example. The Earth is has mean radius of 6,371 km and so 6371 * 3.141 / 180 = 111.2 km/degree. Now if the map is at 49° N, then cos 49° = 0.6506 and that times 111.2 is 72.34 km/degree.
Now, lets say that's too big, but 1/4 that would work. 1/4 of a 60 minute degree is 15 minutes, and 1/4 of 111.2 km x 72.34 km is 27.80 km x 18.09 km.
So, scale that to your map, and draw your grid.
If you want to label it, degrees can be written with the degree sign ( ° ), minutes with the prime sign ( ′ ) or an apostrophe ( ' ) and seconds with the double prime sign ( ″ ) or a double quote ( " ). For instance, 49°32′16″ is 49 degrees, 32 minutes, and 16 seconds.
The specific projection that results from all this is Secant Normal Equidistant Cylindrical. You are probably familiar with Tangent Normal Equidistant Cylindrical (Also known as Platee Carre) which is where the graticule is all squares. The secant form just wraps the map into a tighter cylinder which then cuts through the earth at two parallels, called the "standard parallels", which have the same absolute value (N and S 45° for instance). The tangent form has one standard parallel, the equator. Either way, the map gets more distorted as you get further from the standard parallels, so by putting a standard parallel in the middle of your map, and not going very far away from it, (which is what this tutorial describes) you don't have any significant distortion in your map. There are better projections for this that real world maps would use, but they are more complicated.
Last edited by Hai-Etlik; 12-05-2011 at 03:13 AM.
Reason: Added illustration
You know, I really think this thread should have been picked up a bit more ... the maths is really quite simply and the post is very to the point.
Maybe you should have made a bigger set of equations ... or was it the equations that scared off the commenters?
Last edited by Lukc; 11-20-2011 at 04:53 AM.
I figured anyone who could read mathematical notation would know enough to do the math themselves and so wouldn't need the tutorial. It really is just the most basic of trig.
∆y = r ∆φ
∆x(φ) = r ∆λ cos φ
Where r is the radius of the planet, ∆φ is the angular spacing of the parallels of latitude, ∆λ is the angular spacing of the meridians of longitude, ∆y is the linear spacing of the parallels of latitude, and ∆x(φ) is the linear spacing of the meridians of longitude at a latitude of φ, with all angles expressed in radians.
Thanks for drawing attention to this post, Lukc. Very helpful explanation indeed which even my tiny brain can understand. Thanks Hai!
Thanks, glad you both found it useful.