Which projection is used for small region maps?

[Piotr]

My first thought was Mercator projection, but then I noticed that graticule wasn't straight. From what I heard, I think they use Transverse Mercator at appropriate central meridian. Is it right?

[daan]

That’s right. Most modern national grid systems use the ellipsoidal transverse Mercator, or some minor variation of it.

— daan

[Piotr]

What do you mean by a grid system?

[RogerOwens]

Countries, and sometimes smaller jurisdictions, like U.S. states use a co-ordinate system that refers to points, in that country (or sometimes the whole world).

A plane-coordinate system is a co-ordinate-system consisting of a square grid of straight lines on some flat map of the region.

As daan mentnioned some sort of Transverse-Mercator is the most popular map on which to base a plane-coordinate system.

For example, the U.S. used to use the UTM (Universal Transverse Mercator) plane-coordinate system, based on separate Transverse-Mercator maps for each of many longitude-bands on the Earth. (a newer variation of UTM has since replaced it, I read).

Those Transverse-Mercator maps weren't necessarily actually printed and sold, but the UTM co-ordinate lines drawn on maps (on whatever projection) were the same lines on the Earrth as the square-grid on the base-map, the Transvserse-Mercator map on which the UTM plane co-ordinate system was based.

In fact some maps actually were printed using the very same Transverse-Mercator map that the UTM co-ordinates are based on, so that the UTM grid on the map is really a square-grid.

I read that England used such a system. Its National Ordinance maps (I read) are just pieces of the larger Transverse-Mercator map that is the basis of their plane-coordinate system.

Each U.S. state, too has its own plane-coordinate system, and there are or were some maps printed that used a state's base-projection, so that the state-coordinate grid on the map, then, really is a square coordinate system.

But often a map that has UTM coordinatess and/or state coordinates marked on it, as a grid on the map, or ticks at the map's edges, isn't on the same projection as the plane coordinate system's base-map, and so then the grid lines drawn on the map aren't truly a square grid on that map.

For instance, there are or were a lot of U.S. Geological Survey maps with the UTM grid drawn on the map, and edge-ticks for the state coordinate system, that were on a different projection from the base projections for those two coordinate systems.

Michael Ossipoff

[RogerOwens]

But, as for municipal maps and road-maps, they don't tell the projection (but the British National Ordinance maps, which are road-maps too, are known to be Transverse-Mercator). In this country, though, the road-maps, like the municipal maps, are privately-published, and they aren't labeled for their projection.

I'd like it if they were labeled for their projection, but I guess that they figure that most people wouldn't care. But how much trouble would it be to have, in small-print, at the somewhere at the bottom of the map, a little statement of what projection the map uses?

I don't know what projection(s) they use. It seems to me that daan might have suggested that they likely use the state coordinate system's base-map projection, but I don't remember for sure.

It would seem to make sense, and save money and labor, to compile municipal maps and road-maps from already-existing maps, such as official government maps that cover the whole state.

By the way, it seems to me that I read that, before their adoption of Transverse-Mercator as their national grid-system base-map, England used the Cassini Projection for that purpose. Cassini is a Transverse Cylindrical-Equidistant.

Michael Ossipoff

[Piotr]

I wonder why they used Transverse Mercator and not Mercator, Stereographic, cylindrical equal area on correct standard parallel or equirectangular with correct standard parallel. What's special about transversity?

[RogerOwens]

I wonder why they used Transverse Mercator and not Mercator, Stereographic, cylindrical equal area on correct standard parallel or equirectangular with correct standard parallel. What's special about transversity?

With the Transverse Mercator, a whole longitude-band on the Earth is near the projection's "equator", meaning that it in the region of very little variation in scale.

That makes it possible to assume, with acceptable accuracy, that the map's nominal scale is accurate, and that the plane coordinate system's square grid is really a uniform square grid on the ground.

With ordinary equatorial-aspect Mercator, the U.S., Poland or Germany is a long way from the equator, and therefore is in a region of relatively rapid variation in scale.

Cylindrical equal-area couldn't have an arbitrary standard parallel as its equator, because its equator has to be a great circle.

Anyway, you don't want it to be equal-area or equidistant, because a conformal map is much more useful for navigation, routefinding, surveying, and distance-determination.

Stereographis is used in the UTM system, for the polar regiions, but it doesn't lend itself to dividing the rest of the world up conveniently into zones. The Transverse Mercator uses meridians as the "equator" of its longitude-bands, and that's more convenient for dividing the world into zones.

England of course has more north-south extent, lending itself best to Transverse Mercator.

Michael Ossipoff


Michael Ossipoff

[daan]

I wonder why they used Transverse Mercator and not Mercator, Stereographic, cylindrical equal area on correct standard parallel or equirectangular with correct standard parallel. What's special about transversity?

Transverse Mercator gives a conformal representation with correct scale along a chosen meridian. This is obviously beneficial for regions that are naturally north-south in extent. Beyond that, however, what transverse Mercator enables is an array of zones that map entire regions or the whole world in a systematic way. This is expressed in its ultimate form by the Universal Transverse Mercator (UTM) system. These systems, both national and UTM, have associated “grids”. The grids are rectangular overlays that express coordinates in meters instead of in geodetic coordinates. Because systems are highly regularized, they give engineers, militaries, and the populace at large a convenient way to express locations without having to deal with spherical coordinates.

— daan

[RogerOwens]

One advantage of a plane co-ordinate system is that, because it's made so that you're in the base-map's region of lowest scale-variation, you can often assume, with acceptable accuracy that the co-ordinate system's numbers are a good match for actual distances on the ground,. That makes provides an easy way to estimate the distance and direction of a point, if you know its plane-coordinates (such as UTM co-ordinates, or co-ordinates in a state co-ordinate system), and your own plane co-ordinates.

If rangers get a distress-call that gives the caller's UTM co-ordinates, and the "northing" of those co-ordinates differs from your own "northing" by N, and its easting differs from your easting by E, then you know that you need to travel in a direction such that the tangent of your direction of travel, with respect to grid north, is E/N.

So that just means that, if E/N = .5, to the left of grid-north, then you place the end of a foot ruler two feet away from you, at grid north, and the other end of the foot ruler will be in the direction that you must go to reach the caller.

And the distance you need to travel is the square root of E squared plus N squared.

Michael Ossipoff

[Piotr]

"Cylindrical equal-area couldn't have an arbitrary standard parallel as its equator, because its equator has to be a great circle."

No. What I mean is stretch this projection to make a standard parallel coincide with small region you want to view.

An alternate method that works for all projections with a point without angular distortion is to recenter the map so that the country is on that point.