2.0 Introduction
Maps, whether analog or digital, and spatial data, whether in vector or raster format, are related to some location. We
mostly refer to these locations using coordinate systems. A coordinate system is a set of rules that specifies how
coordinates are assigned to locations.
Three-dimensional spatial coordinate systems are used to locate data on the surface of the Earth. For instance, any
point on the earth can be located by means of spatial geographic coordinates ( , , h ) or geocentric coordinates (x,y,z
Spatial geographic coordinates ( , h )
Spatial cartesian or geocentric coordinates ( x, y, z )
Plane coordinate systems are used to locate data on the map plane. E.g. any point on the map plane can be located
by means of two-dimensional cartesian (or rectangular) coordinates (x,y) or two-dimensional polar coordinates ( ,d ).
Plane rectangular coordinates (x, y)
2D polar coordinates (, d)
2.1 Spatial coordinate systems
2.1.1 Geographic Coordinates
The most widely used global coordinate system consists of lines of geographic latitude and longitude. Lines of equal
latitude are called parallels. They form circles on the surface of the ellipsoid. Lines of equal longitude are called
meridians and they form ellipses (meridian ellipses) on the ellipsoid.
The geographical coordinate system
The latitude of a point P (see figure below) is the angle between the ellipsoidal normal through P' and the equatorial
plane. Latitude is zero on the equator (= 00) and increases towards the two poles to maximum values of  = +90 (N
900) at the North Pole and  = - 90o (S 900) at the South Pole.
The longitude  is the angle between the meridian ellipse which passes through Greenwich and the meridian ellipse
containing the point in question. It is measured in the equatorial plane from the meridian of Greenwich = 00 either
eastwards through  = + 180o (E 1800) or westwards through  = -1800 (W 1800).
Latitude and longitude representing the geographic coordinates ,  of a point P with respect to the selected referenc
surface. They are always given in angular units (e.g. City hall Enschede:  = 520 13' 26.2" N,  = 60 53' 32.1" E).
Spatial geographic coordinates (,, h) are obtained by introducing the ellipsoidal height h to the system. The
ellipsoidal height of a point is the vertical distance of the point in question above the ellipsoid. It is measured in distance
units along the ellipsoidal normal from the point to the ellipsoid surface. The concept can also be applied to a sphere as
the reference surface.
The spatial geographical coordinate system
2.1.2 Geocentric Coordinates (X,Y,Z)
An alternative and often more convenient method of defining a position is with spatial cartesian coordinates. The system
has its origin at the mass-center of the earth with the x and y axes in the plane of the equator. The x-axis passes throug
the meridian of Greenwich, and the z-axis coincides with the earth's axis of rotation. The three axes are mutually
orthogonal and form a right-handed system.
The spatial geocentric coordinate system
It should be noted that the rotational (spin) axis of the earth changes its position with the time (catchword: polar motion
Due to this the mean position of the pole in the year 1903 (based on observations between 1900 and 1905) was used t
define the so-called "Conventional International Origin" (CIO).
2.2 Plane Coordinate Systems
A flat map has only two dimension width (left to right) and length (bottom to top). Transforming the three dimensional
earth body into a two-dimensional map is subject of map projections. Here, like in several other cartographic
applications, two-dimensional coordinates are needed to describe the location of any point in an unambiguous and
unique manner.
2.2.1 Cartesian Coordinates
One possibility of defining a point in a plane is to use plane rectangular coordinates. This is a system of intersecting
perpendicular lines, which contains two principal axes, called the X- and Y--axis. The horizontal axis is usually referred
to as the X-axis and the vertical the Y-axis (Note that the X-axis is sometimes called Easting and the Y-axis Northing).
The intersection of the X- and Y-axis forms the origin. The plane is marked at intervals by equally spaced coordinate
lines.
The 2D cartesian coordinate system
Giving its two numerical coordinates Xp and Yp, one can precisely and objectively specify any location P on the map.
Normally, the coordinates Xp= 0 and Yp = 0 are given to the origin. However, sometimes large positive values are adde
to the origin coordinates. This is to avoid negative values for the X - and Y -coordinates in case the origin of the
coordinate system is located inside the area of interest. The point which has then the coordinates Xp= 0 and Yp= 0 is
called the false origin. Rectangular coordinates are also called cartesian coordinates after Descartes, a French
mathematician of the seventeenth century.
2.2.2 Polar Coordinates
Another possibility of defining a point in a plane is by polar coordinates. This is the distance d from the origin to the poin
concerned and the angle a between a fixed (or zero) direction and the direction to the point.
The 2D polar coordinate system
The angle a is called azimuth or bearing and is counted clockwise. It is given in angular units while the distance d is
expressed in length units. Bearings are always related to a fixed direction (initial bearing) or a datum line. In principle,
this reference line can be chosen freely. However, in practice three different directions are widely in use: True North,
Grid North and Magnetic North. The corresponding bearings are called: true bearing or geodetic bearing, grid bearing
and magnetic or compass bearing.
Polar coordinates are often used in land surveying. For some types of surveying instruments it is advantageous to mak
use of this coordinate system. Especially the development of precise remote distance measurement techniques has led
to the virtually universal preference for the polar coordinate method in detail survey.
2.3 Map grid and graticule
The grid represents lines having constant rectangular coordinates (x, y). The grid is almost always a rectangular syste
and is used on large and medium scale maps to enable detailed calculations and positioning. Plane coordinates and
therefor the grid, are usually not used on small-scale maps, maps smaller than one to a million. The scale distortions
that result from transforming the curved Earth surface to the map plane are so great on small-scale maps that detailed
calculations and positioning are difficult.
The grid and graticule of the Dutch National coordinate system (at small scale)
The graticule represents the projected position of the geographic coordinates at constant intervals, or in other words th
projected position of selected meridians and parallels. The shape of the graticule depends largely on the characteristics
and scale of the map projection used.
The world mapped in the Transverse Mercator projection with a 15 degrees graticule
The grid and graticule spacings on a map vary depending on the scale of the map. E.g. on the 1: 50 000 topographic
map of the Netherlands graticule lines or ticks are shown at every 5 minutes and grid lines at every kilometer.
The map sheet limit or neat line (the line enclosing the mapped area) can either be formed by the outline of the graticu
or the grid. The grid as outline of the map has the advantage of being rectangular, hence the map face of each map
sheet will be exactly the same size. The graticule as outline of the map might give a curved outline, but shows
immediately the extent of the map sheet in the geographical system
2.4 References
Knippers, R.A. (1999). Geometric Aspects of Mapping. Non-published notes, Enschede, ITC.
Mehlbreuer, A. Geometric Fundamentals of Mapping. Non-published notes. Enschede, ITC.