Best projection for preserving parallel lines?
I have some existing lines on my map, and I need to draw lines parallel to them at various distance. GRASS (GIS software I'm using) will draw those lines easily enough, but I want to be sure they're parallel on the globe, not just the map projection. What projection will make parallel lines on the globe parallel on the map? It doesn't need to be a whole-globe projection, I can split into sections if need be.
(Context: the lines I have are ocean ridges. I intend to create a set of lines parallel to them, and assign depths to those lines (ocean floor gets deeper away from the ridges; there's a formula that describes this quite precisely. Then I can interpolate a DEM from those depths.)
There are no such things as parallel lines on a globe.
A "line" on a sphere is a Great Circle, and a "line segment" is an arc of a great circle. Any two distinct great circles intersect exactly twice, and so there are no parallel lines.
We call "lines of latitude" "parallels" but they aren't really, because they better thought of as curves than straight lines. They are "Equidistant curves" or "lesser circles" which correspond to ordinary circles in Euclidean Geometry.
To best plot an equidistant curve, my first thought would be Oblique Equidistant Cylindrical that is tangent along the fault.
Hmmm...and now I realise that the concept of a parallel line, in the geometric sense, isn't even clearly applicable to spherical geometry.
So rather, I think I need to construct a great circle that intersects the ridge at right-angles, then construct further (segments of) great circles at right-angles to that. (This is hypothetical; I won't actually need to go through those steps, I'll just do parallel lines on the map.)
That tells me I need a conformal projection and straight lines on the map need to be as nearly great circles as possible. Distance preservation is less of an issue since I can use the software to find the true geodesic distances.
This points me, I think, to an azimuthal stereographic projection. (Or rather a set of them, each centred on a different area of interest).
EDIT: And while I was writing that post up I see Hai-Etlik has beaten me to it. Maybe his suggestion would work better.
EDIT2: And now I'm reminded of something I overlook before. Over a long distance, the "parallel" line isn't the same distance everywhere from the ridge - for the ridge itself spreads at different rates along its length. However, since I have lots of short segments I can probably treat each piece as an equidistant line.
Last edited by cantab; 07-10-2011 at 02:46 AM.
I've decided the oblique equidistant cylindrical projection is the best for my purposes. Unfortunately, it's also a royal pain to get in GRASS. I've asked on the GRASS mailing list, but maybe someone else here has worked with that projection? (It's really obscure, hence Google has been of little help.)