# Cartographic notes

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# Coordinates and Projections

## Coordinate terminology

Everyone has heard of longitude and latitude (where longitude is west-east and latitude the north-south measure. Try to find a mnemonic you like).

Printed maps and user-friendly display often use a degree-minute-second format such as 35°30'45"W, where the letter specifies on which hemisphere or which side of Greenwich it is.

In computation you usually work with the decimal degree equivalent as it is directly a float value: a simple number that is linearly comparable, and uses positive/negative values for the hemisphere directions. For example, 35°30'45"W is -35.5125 (which is 30 + 30/60 + 45/3600).

(Some physical calculators (and loyal software imitations) that do dd-dms calculations may show 35°30'45" as 35.3045. (Rather important to know to use it correctly.)

A meridian is a line of longitude - a line that runs from north pole to south pole.

A particular meridian is given by the latitude. The prime meridian that runs through greenwich has 0° latitude.

Easting and northing are coordinates in a Cartesian system, which by convention are usually in meters.

The orthogonality that this implies is often assumed, and is only somewhat incorrect when talking about a small area (relative to the sphere it is on).

They are used in some coordinate systems, and also in some projections. For example, the output of UTM is given in easting and northing (see UTM details, though).

## Coordinate system troubles and solutions

Coordinates should never be separated from the description of which coordinate system they are in.

If you just wanted to roughly draw the shape of a country that coordinates describe, you can get away with forgetting the actual meaning of the coordinates. If your country isn't too large, anyway. If accuracy matters in any way, you need a description of the coordinate system that each set of coordinates you use are in.

Omitting it is asking for trouble - there are dozens of coordinate systems superimposed on Earth, and most differ only in minor details you won't notice at a glance, but lead to real inaccuracies at a detailed scale.

There are various things that may make coordinates incompatible, either in a way where the numbers don't directly correspond but can still be fairly accurately converted, or not so much.

For a basic example, different systems take different points on earth as the zero reference. Systems made for a single country often put it just outside the country, and may use their own units - meters, kilometers, miles, degrees, something else. The differences aren't always obvious either; there are two Australian systems that differ (as I recall) a few hundred meters, which is invisible when you view all of Australia, but very obvious once you combine coordinates at city or street level.

A few systems assume the world is a perfect sphere. But most use an ellipsoid formula to be more accurate (consider e.g. that the polar and equatorial radii differ about 20 km). You could dig into the math, or just intuitively believe that drawing on a slightly differently shaped surface means not only a different depth, but also that going by degree coordinates means a slightly different location (consider an exaggerated case, something like an egg), and also that there is no trivial or clearly optimal way to convert between coordinates mapped onto different shapes.

These two and a few other details mean that if you have more than one set of data, you must know what coordinates in each mean to be able to use them in the same map or conclude that you cannot do this conversion at all, and get results that are accurate within some percentage.

GIS programs will usually handle this for you fairly formally, and transparently when they can (although some make it relatively hairy), given that you most tell them the coordinate system for everything, and/or that this is metadata stored along with data you import.

In ESRI's ArcGIS, these are named Geographic Coordinate Systems (GCS), a term I'll use here for consistency's sake. They are proper coordinate systems that consist of:

• cartesian coordinate system (lat-long or other) with:
• a datum, in combination with
• an ellipsoid

Notes:

• in practice, most any datum defintion implies a particular ellipsoid, so as a user you only have to know about the datum.
• Systems with different ellipsoids are not strictly compatible.
• For everyday use, the name of the datum often implies the right GCS, because most of the time a datum implies a specific ellipsoid, and most use the same coordinate system.

### Implied 3D

Technically speaking, GIS programs want and (implicitly) work with geodetic data ("related to the science of measuring the earth").

To grasp what geodetic coordinate systems do, consider that geodetic coordinates exist on an imperfect sphere.

This also explains why different ellipsoids are fundamentally incompatible, at least strictly: if you plunk the same 2D coordinates on a fatter earth, they'll most obviously move in height (which you could choose to ignore in flat maps), but they'll also move slightly sideways (from a purely polar-coordinate view), and this will happen more in some places than in others.

On the scale of a world map it is not perceptible (a few to a few hundred meters of actual distance?(verify)), but for purposes like GPS this would be unacceptable. This is the reason GPS uses a GCS that is accurate to a few centimeters(verify), namely WGS84.

### Datums and ellipsoids

You could say that a complete system that lets you convert/project between cartesian coordinates and geodetic lat-long coordinates.

This can be used to translate coordinates between datums as long as they use the same ellipsoid.

In reality, regions will use different local coordinate systems over time. For example, in Australia there are AGD84 and GDA94 (see e.g. [1]), which are approximately 200m different from each other. North America has seen NAD27, NAD83 (similar to WGS84, actually), and others.

WGS84 is the default for many modern data sets and services. It is more accurate than most, which is why GPS uses it. It is also global (unlike a number of others) making it an obvious choice for most modern gazetteers.

Note that while many datums are based on a particular datum available at the time, a few systems define their own. For example, WGS84 is an improvement over WGS72 mostly in that the ellipsoid was based on somewhat more accurate data about the earth. For this reason, WGS84 can be said to be an ellipsoid as well as being the datum that uses that ellipsoid.

See also:

## Projections, in general

Projection is a general term and something done in different ways in different data. It just mentions one transformation (a projection), but there are more. What GISes do are projections from:

• cartesian to geodetic (knowing the data's datum)
• geodetic to cartesian (knowing the data's datum)
• geodetic to map (with a specific map projection)

Most of this is abstracted out and happens behind the scenes: if your map layers have datums defined, interpreting the data in that layer geodetically, converting between datums and such things will Just Happen, because that's central to working with our data.

That aside, that data has to be projected onto two dimensions before you can see it on a monitor or paper, which is a whole new bag of tricks and problems. This because now the point is arriving at 2D coordinates that you can draw - and you need to destroy some data to do so.

This too is done by projections, just a different set. The simplest map projection is Mercator, which is really just interpreting longitude and latitude as if they were a grid. This will give you strong deformations the further you get from the equator, which is logical when you consider longitudional lines meet at the poles.

There are four major things you value - correct directions (angles), distances, bearings and shapes. Not all can be correct at the same time when in 2D, so you should choose the projection based on what is most important to you. For example, in an oceanfaring map or roadmap, distances and directions are important; in a political or symbolic map shapes may be more interesting. See this map projections list for common map projections and their features.

Notes:

• Arc lets you set the output projection for the entire map, which it calls a Projected Coordinate System, PCS. A simple choice for many tasks is UTM (see below).
• GRASS seems to link the GCS and PCS closer together

## Map projections

### UTM

Universal Transverse Mercator, UTM, is a (WGS84-based(verify)) system around Transverse Mercator. This is based on plain Mercator, but:

• with the equator vertical,
• more or less moves what you want viewed to this equator.

Roughly speaking, it pretends a country is centered on the equator of a sphere, and unwraps from there. The equator is where the least distortion occurs - within about 15° (around 2000 miles, 3300 kilometers) it is hard to discern shape problems, so relatively small countries can be drawn fairly well.

The mentioned centering is around a point, or rather zone: the UTM zone. Zones are 6 degrees wide (numbered) and 8 degrees high (lettered). The zone numbers run from 1 to 60. 1~25 covers the US, 28~60 covers europe and asia. Letters A, B, Y, and Z are used for the polar system, (UTM-)UPS. Letters O and I are not used, leaving 10 letters on each side of the equator. They cover 80 degrees in the southern, and 84 degrees in the northern hemisphere (X is 12 degrees instead of 8). (There are two irregularities to this grid, see below)

For example, the center of germany lies in zone 32 on the northern hemisphere; '32 North' or '32N' is the sort of thing you need to feed into converters.

The output of UTM is easting and northing, in meters. Note that these are relative to a poin tin the zone, so you must know the zone if you want to convert UTM to other types of coordinates.

Beware of converting coordinates individually; half of the point is pretending they are all in the same zone.

UTM's easting is given a false easting of 500,000, meaning that the easting value is relative to the point 500km west of the UTM zone's central meridian. to avoid negative numbers (The range that is actually used is about 167,000 through 833,000).

Northing is relative to the equator (and not the zone). In the northern hemisphere it is simply that (watch datum details, though), while in the southern hemisphere there is a false northing of 10,000,000, again to avoid negative numbers.

Since polar regions (>84 degrees north and >80 degrees south from the equator) are handled separately, northern-hemisphere northing ends at about 9,328,000, and southern-hemisphere northing begins at about 672,000.

There are two irregularities to the UTM zone grid:

• V31 is 3° and V32 is 9° wide, so that the latter accommodates southwest Norway
• Svalbard:
• X33 and X35 are 12° wide each,
• X31 and X37 are 9° wide, and
• X32, X34 and X36 do not exist

The zone maps are probably easier to understand than this description, so look at:

See also:

See:

## Details for

### Arc

ESRI wants a Geographic Coordinate System set per map layer, or it will assume one (which may not be correct correct). ArcGIS programs complain if a map contains data sources with different datums, because there is no way to accurately line up most data mixes.

To choose a coordinate system:

• go to the View Menu,
• select Data Frame Properties,
• the Coordinate System tab,
• go to the bottom tab and choose the coordinate system.

UTM sits under Predefined, Projected Coordinate Systems, UTM, the chosen ellipsoid, and then the chosen UTM zone. You may want to add it to favourites if you want to use it again, it's less clicking next time.

ArcGIS shape files (.shp) should have an according .prj file that stores the GCS (though inpractice it is optional).

If you know the the GCS (it may be mentioned in text files that came with the data / shape file), add it. If not, you'll have to guess - though it won't be perfect, you may guess better than the default. Do clearly mark that you did this, otherwise this data will confuse you or others later.

One way to create the .prj file is to:

• run ArcCatalog (this is apparently also possible via the ArcToolbox program or snap-in, but the exact instructions differ)
• locate the file,
• right-click, click Properties,
• click the Fields tab,
• select the Shape row;
• the properties show up, one of which is called Spatial Reference; click the edit (...) button, and you are given a cordinate system properties tab by clicking Select and the finding and and applying one.

# On digital shapes

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, or tell me)

# On file formats

## ESRI shapefiles

Actually refers to a set of files with the same name and different extensions. Such a set should include at least:

• .shp (geometry; binary format)
• .shx (index into the .shp; binary format, mostly a simple list of (offset,length) pairs)
• .dbf (attributes, in dBase III format)

The order of features in each of these matters, as implicit feature numbering.

There are over a dozen optional files including:

• .prj (projection/coordinate system; plain text)
• various documented and undocumented files, most of them indices of some sort
• .cpg - to specify the code page used in the .dbf (verify)

See also:

## File formats

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, or tell me)

Libraries:

# Data sources

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, or tell me)

## Locations

Coordinates are either collected by you though GPS measurement (note this is usually accurate to 10-20meters, sometimes better, sometimes worse) or looked up, usually on the internet. Some interesting sites are:

• GeoNames
• also interrelated with other data such as country, population
• available via web API, also as downloadable data

See also:

(Side note: If you use Arc: ArcToolbox (a program and snap-in for Arc programs) can do a bunch of file conversions. You can usually also find some way of doing it without Arc, though they are regularly more complex)

## Shapes

...such as rivers, points sets, and (segmented parts of) border polygons such

Indexes and relatively free data:

• GSHHS (Global Self-consistent, Hierarchical, High-resolution Shoreline database) [4] [5]
• consists of closed polygons, and only shorelines, no political borders.
• Generic Mapping Tools (GMT) [6]
• has state borders, country borders, and river line data
• and GSHHS boundaries (separately)
• matplotlib's basemap [7]
• state, country, river data from GMT
• Shoreline data from GSHHS (separately)
• country boundaries and continent/ocean lines are separate, so country boundaries do not directly exist.
• st_cntry_bndrs.tar.gz (a file that hangs around a few places)
• country boundaries (created from GSHHS shorlines + (which?(verify)) political boundaries)

• Digital Charts of the World / VMAP
• cartographic data, attribute data
• originally created by/for the US Defense Mapping Agency (for pilot navigation needs)
• created by digitizing Operational Navigation Charts. Jet Navigation Charts. {{comment|(Note that there are some inaccuracies due to visibility decisions -- but at a fairly small scale)||
• now available through NIMA
• Licensing:
• The ARC/INFO version is copyright by ESRI.
• The VPF format is public domain (verify)
• (am not quite sure about the editions and such)
• See also notes and links below

• Libre_Map_Project
• WDB-II (CIA World DataBank II)
• Micro-WDB-II, available at NGDC and other places

Data tending to the proprietary:

• Geography Network, related to ESRI, is a generally searchable index linking to many smaller websites, including government and commerical sources.
• The ESRI DataDownloader allows you to select some small parts of the world, layer types, and supplies this data as shape files (+GCS).
• The ESRI map CD(s).

Unsorted

• World Vector Shoreline (WVS)

See also:

## Elevation

• GTOPO30: free worldwide DEM data from USGS, at 30-arc second resolution (~1km)

• Shuttle Radar Topography Mission (SRTM)
a research effort that yielded partly one-second, partly three-second (~30m and ~90m) DEM data
covering most of the world (56°S to 60°N, which excludes the poles, Iceland, most of Greendland and, Canada, Norway, part of Russia, etc.).
Freely available(verify)
https://www2.jpl.nasa.gov/srtm/

## Semi-sorted (data sources)

### Notes on VMAP / Digital Charts of the World (DCW)

 This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, or tell me)

(Note that some parts are quite unverified)

Derived from paper maps that were quite detailed, but also somewhat inaccurate when, at the time of the paper maps' creation, slight tweaks suited visibility for end use.

Georeference is apparently WGS84, with elevation based on MSL (Mean Sea Level) [8].

VMap0 and 1 are in VFP format; see US Military Standard MIL-STD-2407, and this list of related MIL standards.

You will probably want to find a converter to whatever format you want to really use.

VMap is divided into:

• Vector Map Level 0 a.k.a. VMap0, freely available, 1:1,000,000 level detail
• Public domain, publicly available, e.g. from the NGA in four files (v0noa.tar.gz: North America (NOAMER), v0soa.tar.gz: South America, Africa (and Antarctica, SOAMAFR?), v0eur.tar.gz: Europe (and north asia, EURNASIA), v0sas.tar.gz: Asia and Asutralia/Oceania (SASAUS))
• Vector Map Level 1 a.k.a. VMap1, partly available, 1:250,000 level detail
• VMAP1 is divided somwhat irregularly into 234 geographic zones (originally) distributed on CDs (see e.g. [9]), 55 or 57(verify) of which are public domain and downloadable (e.g. from NGA) and other places, the rest have to be bought. (The filenames look like v1125.tar.gz, where 'v1' refers to VMAP Level 1, and the 125 refers to area 125)
• a subset is publicly available
• license?
• Vector Map Level 2 a.k.a. VMap2. (derived from TLM) (1:100,000 and 1:50,000)
• license?
• Urban Vector Map a.k.a. UVMap (derived from city graphics)
• license?

Editions:

• Edition 1: in 1992, by DMA, referred to as DCW, a product discontinued in 1997.
• Edition 2 (1995), VMap (which practically replaced DCW as a product in the same year(verify))
• Edition 3 (1998), VMap

(Re)publications:

• DMA? (original, public?(verify))
• NGA?
• Arc (data in Arc format is copyrighted to ESRI)
• manifold.net (free?)

Links (introductions, other notes and relevant information):

Integration

# Glossary

## Map/data types types

• Bathymetry/Bathymetric: Bottom of lakes, oceans, etc., often in relief or iso contour form, e.g. coastal reliefs
• DEM: Digital Elevation Model, often in raster (cell data, can be images) or triangulated irregular network form
• DTM: Digital Terrain Model, another name for DEM
• DSM: Digital Surface Model: Like a DEM, but also includes things on the surface
• DLG: Digital Line Graph (USGS term?)
• DRG: Digital Raster Graphics (USGS term?)
• Hypsometry: Measurement of elevation above sea level. For the most part, see DEM and such.
• GCS: Geographic Coordinate System (Arc / ESRI term)
• PCS: Projection Coordinate System (Arc / ESRI term)
• Topography: Surface detail

## Geodetics & projection

• Datum
• Geodetic, Geodesy: related to the science of measuring the earth
• NAD (datum): North American Datum (There are several, e.g. NAD83)
• Reference ellipsoid: "a mathematically defined surface that approximates the true figure of the Earth or geoid."

### Projections

Include...

• UTM: Universal Transverse Mercator (map projection)
• Equirectangular (also equidistant cylindrical, plate carré, carte parallelogrammatique projection (CPP))

Cut-ups:

Unsorted

• Aitoff-Wagner
• Larrivée

## Organizations

• DMA: US Defense Mapping Agency, merged (in various steps) into NIMA
• NGA: National Geospatial-Intelligence (previously NIMA) [14]
• NGDC: National Geophysical Data Center [15]
• NIMA: National Imagery and Mapping Agency (now NGA)
• USGS: United States Geological Survey

# Unsorted

On the scale/accuracy of coordinates:

• a degree is roughly ~111 km
• a minute is roughly ~1.85 km
• a second is roughly ~31 m

...but note this only applies directly to latitude; longitude is 111km at the equator, 0km at the poles, and approximated by 111*cos(latitude__probably_in_radians) inbetween.

for a more accurate approximation (still imperfect as it still assumes a perfect sphere) you want the Great-circle distance formula[16]:

```r * acos(
sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(lon2 - lon1)
)
```

...where r is the radius. The value you choose depends on/controls the units of the result, e.g. 6378.7 for km, 3437.7 for nautical miles, 3963 for statute miles.