Koordinaattijärjestelmät¶

Yleistä¶

Karttaprojektiot yrittävät kuvata Maapalloa tai sen osaa tasolle, kuten paperille tai tietokoneen näytölle. Käytettävä koordinaattijärjestelmä ja sen mukaiset koordinaatit kertovat, mihin kartalla esitettävä kohde todellisuudessa sijoittuu. Käytettävän karttaprojektion ja koordinaattijärjestelmän valinta riippuu esitettävän alueen koosta, sijainnista Maapallolla ja kartan käyttötarkoituksesta.

Karttaprojektioista¶

Perinteinen tapa esittää Maata on pallokartta. Pallokartta säilyttää hyvin laajojen kohteiden muodot, suunnat sekä suhteelliset pinta-alat ja etäisyydet. Sellaista on kuitenkin erittäin vaikea kuljetella mukanaan, eikä se sovellu kuin erittäin pienimittakaavaisille (esim. 1:100 milj.) kartoille.

Pääosa teemakartoista laaditaan huomattavasti suurimittakaavaisemmiksi. Tyypillinen paikkatietoaineisto on kerätty mittakaavaan 1:250 000 tai jopa suuremmaksi esitettävien yksityiskohtien määrästä riippuen. Tämänkokoinen karttapallo olisi vaikeaa ja kallista tuottaa ja lähes mahdotonta kuljettaa. Niinpä kartografit ovat kehittäneet lukuisia matemaattisia karttaprojektioita, joilla voidaan esittää kolmiuloitteinen, lähes pallonmuotoinen Maa kaksiuloitteisella tasolla riittävän hyvin.

Läheltä katsottuna maapallon pinta näyttää varsin pääpiirteittäin tasaiselta. Kun Maata katsoo avaruudesta käsin huomataan sen olevan lähes pallonmuotoinen. Kartat ovat, kuten kartan tuottamista koskevassa luvussa todetaan, todellisuuden kuvauksia. Ne eivät pelkästään näytä kohteita, vaan myös niiden muodot ja keskinäiset sijainnit. Jokaisella karttaprojektiolla on hyvät ja huonot puolensa. Paras projektio kullekin kartalle riippuu kartan mittakaavasta ja käyttötarkoituksesta. Joku projektio saattaa vääristää tolkuttomasti, jos sillä yritetään esittää kokonaista maanosaa, mutta sama projektio sopii erittäin hyvin suurimittakaavaiseen (yksityiskohtaiseen) karttaan yhdestä valtiosta. Projektion ominaisuudet vaikuttavat kartan suunnitteluun, sillä jotkut projektiot soveltuvat suppeiden alueiden karttoihin, toiset itä-länsisuuntaisten ja kolmannet pohjois-eteläsuuntaisten kohdealueiden esittämiseen.

Karttaprojektioiden kolme ryhmää¶

The process of creating map projections can be visualised by positioning a light source inside a transparent globe on which opaque earth features are placed. Then project the feature outlines onto a two-dimensional flat piece of paper. Different ways of projecting can be produced by surrounding the globe in a cylindrical fashion, as a cone, or even as a flat surface. Each of these methods produces what is called a map projection family. Therefore, there is a family of planar projections, a family of cylindrical projections, and another called conical projections (see figure_projection_families)

Figure Projection Families 1:

The three families of map projections. They can be represented by a) cylindrical projections, b) conical projections or c) planar projections.

Today, of course, the process of projecting the spherical earth onto a flat piece of paper is done using the mathematical principles of geometry and trigonometry. This recreates the physical projection of light through the globe.

Karttaprojektioiden virheellisyydet¶

Karttaprojektiot eivät voi koskaan olla täysin virheettömiä pallomaisen pinnan esityksiä tasolla. Projisoinnin tuloksena kartalla voidaan havaita vääristymiä suuntakulmissa, etäisyyksissä ja/tai pinta-aloissa. Joissain karttaprojektioissa esiintyy näitä kaikkia vääristymiä, mutta vain siedettävissä määrin. Esimerkkejä tällaisista kompromissiprojektioista ovat Winkel-Tripel ja Robinson (see figure_robinson_projection), joita käytetään usein maailmankartoissa.

Figure Robinson Projection 1:

The Robinson projection is a compromise where distortions of area, angular conformity and distance are acceptable.

It is usually impossible to preserve all characteristics at the same time in a map projection. This means that when you want to carry out accurate analytical operations, you need to use a map projection that provides the best characteristics for your analyses. For example, if you need to measure distances on your map, you should try to use a map projection for your data that provides high accuracy for distances.

Map projections with angular conformity¶

When working with a globe, the main directions of the compass rose (North, East, South and West) will always occur at 90 degrees to one another. In other words, East will always occur at a 90 degree angle to North. Maintaining correct angular properties can be preserved on a map projection as well. A map projection that retains this property of angular conformity is called a conformal or orthomorphic projection.

These projections are used when the preservation of angular relationships is important. They are commonly used for navigational or meteorological tasks. It is important to remember that maintaining true angles on a map is difficult for large areas and should be attempted only for small portions of the earth. The conformal type of projection results in distortions of areas, meaning that if area measurements are made on the map, they will be incorrect. The larger the area the less accurate the area measurements will be. Examples are the Mercator projection (as shown in figure_mercator_projection) and the Lambert Conformal Conic projection. The U.S. Geological Survey uses a conformal projection for many of its topographic maps.

Figure Mercator Projection 1:

The Mercator projection, for example, is used where angular relationships are important, but the relationship of areas are distorted.

Map projections with equal distance¶

If your goal in projecting a map is to accurately measure distances, you should select a projection that is designed to preserve distances well. Such projections, called equidistant projections, require that the scale of the map is kept constant. A map is equidistant when it correctly represents distances from the centre of the projection to any other place on the map. Equidistant projections maintain accurate distances from the centre of the projection or along given lines. These projections are used for radio and seismic mapping, and for navigation. The Plate Carree Equidistant Cylindrical (see figure_plate_caree_projection) and the Equirectangular projection are two good examples of equidistant projections. The Azimuthal Equidistant projection is the projection used for the emblem of the United Nations (see figure_azimuthal_equidistant_projection).

Figure Plate Carree Projection 1:

The Plate Carree Equidistant Cylindrical projection, for example, is used when accurate distance measurement is important.

Figure Azimuthal Equidistant Projection 1:

The United Nations Logo uses the Azimuthal Equidistant projection.

Projections with equal areas¶

When a map portrays areas over the entire map, so that all mapped areas have the same proportional relationship to the areas on the Earth that they represent, the map is an equal area map. In practice, general reference and educational maps most often require the use of equal area projections. As the name implies, these maps are best used when calculations of area are the dominant calculations you will perform. If, for example, you are trying to analyse a particular area in your town to find out whether it is large enough for a new shopping mall, equal area projections are the best choice. On the one hand, the larger the area you are analysing, the more precise your area measures will be, if you use an equal area projection rather than another type. On the other hand, an equal area projection results in** distortions of angular conformity** when dealing with large areas. Small areas will be far less prone to having their angles distorted when you use an equal area projection. Alber’s equal area, Lambert’s equal area and Mollweide Equal Area Cylindrical projections (shown in figure_mollweide_equal_area_projection) are types of equal area projections that are often encountered in GIS work.

Figure Mollweide Equal Area Projection 1:

The Mollweide Equal Area Cylindrical projection, for example, ensures that all mapped areas have the same proportional relationship to the areas on the Earth.

Keep in mind that map projection is a very complex topic. There are hundreds of different projections available world wide each trying to portray a certain portion of the earth’s surface as faithfully as possible on a flat piece of paper. In reality, the choice of which projection to use, will often be made for you. Most countries have commonly used projections and when data is exchanged people will follow the national trend.

Koordinaattijärjestelmistä tarkemmin¶

Koordinaattijärjestelmien avulla mikä tahansa sijainti maapallolla voidaan yksilöidä kolmen luvun eli koordinaatin yhdistelmänä. Yleisimmin käytettävät koordinaattijärjestelmät voidaan jakaa maantieteellisiin koordinaattijärjestelmiin ja tasokoordinaattijärjestelmiin.

Maantieteelliset koordinaattijärjestelmät¶

Maantieteellisiä koordinaattijärjestelmiä käytetään yleisesti. Niissä sijainti ilmoitetaan leveys- ja pituusasteiden avulla, ja joskus käytetään myös korkeuskoordinaattia. Maailmanlaajuistesti yleisin niistä on WGS84.

Leveyspiirit kulkevat Päiväntasaajan suuntaisesti ja jakavat Maan pohjois-eteläsuunnassa 180:een yhden asteen levyiseen vyöhykkeeseen, joista 90 on kummallakin pallonpuoliskolla. Päiväntasaaja on leveyspiiri 0, ja navat sijaitsevat leveyspiireillä 90 astetta. Sekaannusten välttämiseksi eteläisen pallonpuoliskon leveysasteet merkitään usein negatiivisina (0...-90°), ja pohjoisen pallonpuoliskon leveysasteet vastaavasti positiivisina. Riippumatta sijainnista maapallolla etäisyys leveyspiirien välillä on kutakuinkin yhtä suuri (60 mpk eli ~110 km).

Figure Geographic CRS 1:

Geographic coordinate system with lines of latitude parallel to the equator and lines of longitude with the prime meridian through Greenwich.

Lines of longitude, on the other hand, do not stand up so well to the standard of uniformity. Lines of longitude run perpendicular to the equator and converge at the poles. The reference line for longitude (the prime meridian) runs from the North pole to the South pole through Greenwich, England. Subsequent lines of longitude are measured from zero to 180 degrees East or West of the prime meridian. Note that values West of the prime meridian are assigned negative values for use in digital mapping applications. See figure_geographic_crs for a pictorial view.

At the equator, and only at the equator, the distance represented by one line of longitude is equal to the distance represented by one degree of latitude. As you move towards the poles, the distance between lines of longitude becomes progressively less, until, at the exact location of the pole, all 360° of longitude are represented by a single point that you could put your finger on (you probably would want to wear gloves though). Using the geographic coordinate system, we have a grid of lines dividing the earth into squares that cover approximately 12363.365 square kilometres at the equator — a good start, but not very useful for determining the location of anything within that square.

To be truly useful, a map grid must be divided into small enough sections so that they can be used to describe (with an acceptable level of accuracy) the location of a point on the map. To accomplish this, degrees are divided into minutes (') and seconds ("). There are sixty minutes in a degree, and sixty seconds in a minute (3600 seconds in a degree). So, at the equator, one second of latitude or longitude = 30.87624 meters.

Projected coordinate reference systems¶

A two-dimensional coordinate reference system is commonly defined by two axes. At right angles to each other, they form a so called XY-plane (see figure_projected_crs on the left side). The horizontal axis is normally labelled X, and the vertical axis is normally labelled Y. In a three-dimensional coordinate reference system, another axis, normally labelled Z, is added. It is also at right angles to the X and Y axes. The Z axis provides the third dimension of space (see figure_projected_crs on the right side). Every point that is expressed in spherical coordinates can be expressed as an X Y Z coordinate.

Figure Projected CRS 1:

Two and three dimensional coordinate reference systems.

A projected coordinate reference system in the southern hemisphere (south of the equator) normally has its origin on the equator at a specific Longitude. This means that the Y-values increase southwards and the X-values increase to the West. In the northern hemisphere (north of the equator) the origin is also the equator at a specific Longitude. However, now the Y-values increase northwards and the X-values increase to the East. In the following section, we describe a projected coordinate reference system, called Universal Transverse Mercator (UTM) often used for South Africa.

Universal Transverse Mercator (UTM) CRS in detail¶

The Universal Transverse Mercator (UTM) coordinate reference system has its origin on the equator at a specific Longitude. Now the Y-values increase southwards and the X-values increase to the West. The UTM CRS is a global map projection. This means, it is generally used all over the world. But as already described in the section ‘accuracy of map projections’ above, the larger the area (for example South Africa) the more distortion of angular conformity, distance and area occur. To avoid too much distortion, the world is divided into 60 equal zones that are all 6 degrees wide in longitude from East to West. The UTM zones are numbered 1 to 60, starting at the international date line (zone 1 at 180 degrees West longitude) and progressing East back to the international date line (zone 60 at 180 degrees East longitude) as shown in figure_utm_zones.

Figure UTM Zones 1:

The Universal Transverse Mercator zones. For South Africa UTM zones 33S, 34S, 35S, and 36S are used.

As you can see in figure_utm_zones and figure_utm_for_sa, South Africa is covered by four UTM zones to minimize distortion. The zones are called UTM 33S, UTM 34S, UTM 35S and UTM 36S. The S after the zone means that the UTM zones are located south of the equator.

Figure UTM for South Africa 1:

UTM zones 33S, 34S, 35S, and 36S with their central longitudes (meridians) used to project South Africa with high accuracy. The red cross shows an Area of Interest (AOI).

Say, for example, that we want to define a two-dimensional coordinate within the Area of Interest (AOI) marked with a red cross in figure_utm_for_sa. You can see, that the area is located within the UTM zone 35S. This means, to minimize distortion and to get accurate analysis results, we should use UTM zone 35S as the coordinate reference system.

The position of a coordinate in UTM south of the equator must be indicated with the zone number (35) and with its northing (y) value and easting (x) value in meters. The northing value is the distance of the position from the equator in meters. The easting value is the distance from the central meridian (longitude) of the used UTM zone. For UTM zone 35S it is 27 degrees East as shown in figure_utm_for_sa. Furthermore, because we are south of the equator and negative values are not allowed in the UTM coordinate reference system, we have to add a so called false northing value of 10,000,000 m to the northing (y) value and a false easting value of 500,000 m to the easting (x) value. This sounds difficult, so, we will do an example that shows you how to find the correct UTM 35S coordinate for the Area of Interest.

On-The-Fly Projection¶

As you can probably imagine, there might be a situation where the data you want to use in a GIS are projected in different coordinate reference systems. For example, you might get a vector layer showing the boundaries of South Africa projected in UTM 35S and another vector layer with point information about rainfall provided in the geographic coordinate system WGS 84. In GIS these two vector layers are placed in totally different areas of the map window, because they have different projections.

To solve this problem, many GIS include a functionality called on-the-fly projection. It means, that you can define a certain projection when you start the GIS and all layers that you then load, no matter what coordinate reference system they have, will be automatically displayed in the projection you defined. This functionality allows you to overlay layers within the map window of your GIS, even though they may be in different reference systems.

Common problems / things to be aware of¶

The topic map projection is very complex and even professionals who have studied geography, geodetics or any other GIS related science, often have problems with the correct definition of map projections and coordinate reference systems. Usually when you work with GIS, you already have projected data to start with. In most cases these data will be projected in a certain CRS, so you don’t have to create a new CRS or even re project the data from one CRS to another. That said, it is always useful to have an idea about what map projection and CRS means.

What have we learned?¶

Let’s wrap up what we covered in this worksheet:

• Map projections portray the surface of the earth on a two-dimensional, flat piece of paper or computer screen.
• There are global map projections, but most map projections are created and optimized to project smaller areas of the earth’s surface.
• Map projections are never absolutely accurate representations of the spherical earth. They show distortions of angular conformity, distance and area. It is impossible to preserve all these characteristics at the same time in a map projection.
• A Coordinate reference system (CRS) defines, with the help of coordinates, how the two-dimensional, projected map is related to real locations on the earth.
• There are two different types of coordinate reference systems: Geographic Coordinate Systems and Projected Coordinate Systems.
• On the Fly projection is a functionality in GIS that allows us to overlay layers, even if they are projected in different coordinate reference systems.

Now you try!¶

Here are some ideas for you to try with your learners:

• Start QGIS and load two layers of the same area but with different projections and let your pupils find the coordinates of several places on the two layers. You can show them that it is not possible to overlay the two layers. Then define the coordinate reference system as Geographic/WGS 84 inside the Project Properties dialog and activate the checkbox Enable on-the-fly CRS transformation. Load the two layers of the same area again and let your pupils see how on-the-fly projection works.
• You can open the Project Properties dialog in QGIS and show your pupils the many different Coordinate Reference Systems so they get an idea of the complexity of this topic. With ‘on-the-fly’ CRS transformation enabled you can select different CRS to display the same layer in different projections.

Something to think about¶

If you don’t have a computer available, you can show your pupils the principles of the three map projection families. Get a globe and paper and demonstrate how cylindrical, conical and planar projections work in general. With the help of a transparency sheet you can draw a two-dimensional coordinate reference system showing X axes and Y axes. Then, let your pupils define coordinates (x and y values) for different places.

Further reading¶

Books:

• Chang, Kang-Tsung (2006). Introduction to Geographic Information Systems. 3rd Edition. McGraw Hill. ISBN: 0070658986
• DeMers, Michael N. (2005). Fundamentals of Geographic Information Systems. 3rd Edition. Wiley. ISBN: 9814126195
• Galati, Stephen R. (2006): Geographic Information Systems Demystified. Artech House Inc. ISBN: 158053533X

Verkkosivustoja:

• http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj_f.html
• http://geology.isu.edu/geostac/Field_Exercise/topomaps/index.htm

The QGIS User Guide also has more detailed information on working with map projections in QGIS.

Mitä seuraavaksi?¶

Seuraavassa osiossa katsomme tarkemmin kartantuotantoa.