8.0 Introduction
This section intends to answer your
questions concerning *Geometric Aspects of Mapping*. Answers
are given on a number of commonly asked questions related to coordinate
systems, reference surfaces, map projections, and coordinate
transformations.
8.1 Ask a question
Questions concerning *Geometric Aspects of
Mapping* can be sent to us by filling in the Question Form:
8.2 FAQ on coordinate systems
1. **What is a coordinate system? **A method to locate the position
of objects in two or three dimensions into correct relationship with respect
to each other.
2. **What kind of coordinate systems
are used in mapping?** Spatial coordinate systems (also known as global coordinate systems) are used to locate data either on the Earth’s surface in a 3D space, or on the Earth’s reference surface (ellipsoid or sphere) in a 2D space. Examples are the geographic coordinates in a 2D or 3D space and the geocentric coordinates, also known as 3D Cartesian coordinates. Planar coordinate systems on the other hand are used to locate data on the flat surface of the map in a 2D space. Examples are the 2D Cartesian coordinates and the 2D polar coordinates. Refer to section 2 for details.
3. **What is a graticule? **The graticule represents the projected position of the geographic coordinates (f,l) on a map at constant intervals, or in other words the projected position of selected meridians and parallels. The shape of the graticule depends largely on the characteristics of the map projection used and the scale of the map.
4. **What is a grid? **The map grid represents lines having constant 2D Cartesian coordinates (*x*,*y*). It is almost always a rectangular system and is used on large and medium scale maps to enable detailed calculations and positioning. The map grid is usually not shown on small scale maps (about one to a million or smaller). Scale distortions that result from transforming the Earth’s curved surface to the map plane are so great on small-scale maps that detailed calculations and positioning are difficult.
8.3 FAQ on reference surfaces
5. **What kind of reference surfaces
are used in mapping?** In mapping different surfaces or Earth figures
are used. These include a geometric or mathematical reference surface,
the ellipsoid or the sphere, for measuring locations, and an equipotential
surface called the Geoid or vertical datum for measuring heights. See section 3 for details.
6. **Can we approximate the Geoid with a mathematical model? **The Geoid is extremely undulated due to the large vertical variations between mountains and valleys and mass anomalies. This makes it impossible to approximate the shape of the Geoid with any reasonably simple mathematical model. You also have to realize that the Geoid is a physical surface. We can only measure it, but not describe it mathematically. Another difficulty is that there are several realizations of the Geoid due to local phenomena such as ocean currents, tides, coastal winds, water temperature and salinity at the location of the tide-gauge.
7. **What is a vertical datum?** The
vertical datum, an approximation of the Geoid, is defined as natural reference
surface for land surveying. A vertical datum fits the mean sea level surface
throughout the area of interest and provides the surface to which height
ground control measurements are referred.
8. **Why are there several vertical datums defined? **Historically, the vertical datum (or height datum) is fixed locally not globally by establishing a local mean sea level. Several of these local vertical datums exist in order to minimize distortions due to ocean current, water/sea tides, winds, water temperature, and salinity at the place of tidal gauge. These local measurements of the mean sea level height (zero height) are an approximation of the Geoid. Nowadays there are possibilities to measure the Geoid with gravity satellite missions. They might lead in the future to the establishment of one global Geoid (height datum) with centimeter level accuracies.
9. **Why do we need an ellipsoid as reference surface in mapping?** The physical surface of the Earth is a complex shape. In order to represent
it on a plane it is necessary to move from the physical surface to a mathematical
one, close to the former.
10. **When can we use the sphere as the Earth figure?** The surface of
the Earth may be taken mathematically as a sphere instead of ellipsoid for
maps at smaller scales. In practice, maps at scale 1:5,000,000 or smaller can use the mathematically simpler sphere without the risk of large distortions. At larger scales, the more complicated mathematics of ellipsoids are needed to prevent these distortions in the map. A sphere can be derived from the certain ellipsoid corresponding either
to the semi-major or semi-minor axis, or average of both axes or can have
equal volume or equal surface than the ellipsoid.
11. **What is a horizontal (or geodetic) datum?** A horizontal (or geodetic) datum is defined by the size and shape of an
ellipsoid as well as several known positions on the physical surface at
which latitude and longitude measured on that ellipsoid are known to fix
the position of the ellipsoid. It is used for geodetic control and as reference surface for the planimetric (horizontal) measurements on the Earth surface.
12. **What is the difference between an ellipsoid and a horizontal (or geodetic) datum?** A horizontal datum is the realization of an ellipsoid with a certain position and orientation by means of survey methods. E.g. a local geodetic datum is determined by the dimensions of the ellipsoid, adopted coordinates of one fundamental point and the orientation of the ellipsoid (the azimuth from the fundamental point to another point). A local horizontal datum is determined by the dimensions of the ellipsoid (*a*,*b*), adopted coordinates (geographic coordinates) of one fundamental point (a point of the national triangulation network), and the orientation of the ellipsoid (the azimuth from the fundamental point to another point).
13. **Why are there several ellipsoids
and datums defined?** An ellipsoid and a horizontal datum serve as geometric models
of the Earth surface. Different local ellipsoids with varying position and orientation had to be adopted to best fit the Earth's surface over an area of local interest. This is important to minimize distortions
on maps. A large number of reference ellipsoids and many more local datums
may be encountered in world mapping. Some well-known ellipsoids are
the International, Krasovsky, Bessel, Clark 1880, GRS80 and the WGS84 ellipsoid.
Local ellipsoids serve as reference only for a local area of the Earth's
surface. Global ellipsoids (e.g. WGS84) serve as mean reference for the
entire Earth surface.
14. **What is the difference between a local and global horizontal datum? **A local datum is determined by the dimensions of the ellipsoid (*a*,*b*), adopted coordinates (geographic coordinates) of one fundamental point (a point of the national triangulation network), and the orientation of the ellipsoid (the azimuth from the fundamental point to another point). Positions in terms of the locally adopted datum are propagated through the country, with varying degrees of accuracy, by survey networks. In the past local datums were generally based on one or more astronomically determined positions and the best model of the Earth for the local area. Global (or geocentric) datums (e.g. ITRS, WGS84, ETRS89) are determined by means of satellite-based positioning techniques. By observing satellites one can pinpoint the centre of mass of the Earth, and hence the origin of the geocentric datum. Permanently operating satellite positioning equipment (e.g. GPS receivers) allow the implementation of a global datum (e.g. ITRF2000).
15. **What is a Terrestrial Reference System? **A Terrestrial Reference System (TRS) is a spatial reference system co-rotating with the Earth in its diurnal motion in space. In such a system, positions of points anchored on the Earth solid surface have coordinates which undergo only small variations with time, due to geophysical effects (tectonic or tidal deformations).
16. **What is a Terrestrial Reference Frame?** A Terrestrial Reference Frame (TRF) is a set of physical points with precisely determined coordinates in a specific coordinate system (cartesian, geographic, mapping...) attached to a Terrestrial Reference System. Such a TRF is said to be a realization of the TRS.
17. **What is the International Terrestrial Reference System? **The International Terrestrial Reference System (ITRS) is a global datum for the entire world defined by a 3D geocentric coordinate system. Its origin is located in the centre of mass of the Earth. The* X*-axis is oriented towards the Greenwich meridian, and is orthogonal to the *Z*-axis and the *Y*-axis. The ITRS is realized/established through the ITRF, a distributed set of ground control stations that measure their position continuously (using GPS). Constant re-measuring is needed because of the continuously involvement of new control stations and the tectonic plate motion that causes positional differences in time. This results in a more than one time realization of the ITRS. Examples are the ITRF96 or the ITRF2000.
18. **What is WGS84?** WGS84 is one
of the World Geodetic Systems which provides the basic reference frame
and geometric figure for the Earth. WGS84 provides a positional relation
of various local geodetic systems to an Earth-centered, Earth-fixed coordinate
system, through reports of the DMA of U.S. Defense (D.O.D). GPS measurements
use the WGS84 as reference surface for their measurements.
19. **What is Mean Sea level?** Refer to extern link: <http://www.pol.ac.uk/psmsl/puscience>.
20. **How do we determine the geodetic latitude (***f*) and longitude (*l*) of a point? Satellite based measurements (e.g. GPS) are nowadays used to precisely determine the geodetic coordinates of a point. Before the introduction of GPS the latitude and longitude of a point was measured using geodetic positioning techniques such as triangulation and trilateration, and astronomic observations.
8.4 FAQ on map projections
21. **What is a map projection? **A map projection is a mathematically described technique of how to represent the Earth’s curved surface on a flat map. To represent parts of the surface of the Earth on a flat paper map or on a computer screen, the curved horizontal reference surface must be mapped onto the 2D mapping plane. The reference surface for large-scale mapping is usually an oblate ellipsoid, and for small-scale mapping, a sphere. Mapping onto a 2D mapping plane means transforming each point on the reference surface with geographic coordinates (f,l) to a set of Cartesian coordinates (*x*,*y*) representing positions on the map plane.
*(x*,* y*) _{map projection} = f (f, l)
The corresponding inverse mapping equation transforms mathematically the planar Cartesian coordinates (x,y) of a point on the map plane to a set of geographic coordinates (f, l) on the curved reference surface. See section 4 for details.
22. **What are map projection parameters?** Map projection parameters are part of the projection equations. Common parameters are: *R*=radius of the sphere; *a*=equatorial radius
or semi-major axes of the ellipsoid of reference; *b*=polar radius or
semi-minor axes of the ellipsoid of reference; *e*=eccentricity of the
ellipsoid; *f*=flattening of the ellipsoid; *h*_{o}=scale factor at central
meridian; h=relative scale factor along a meridian of longitude; *k*_{o}=scale factor at standard parallel(s); k=relative scale factor along
a parallel of latitude; l_{o}=central meridian
or longitude of origin; j_{o}=latitude of origin;
*X*_{o}=false Easting; *Y*_{o}=false Northing. Projection
parameters have a significant role in defining a map coordinate system. An extended list of parameters is given in '*Map Projections - A Working manual*', p.viii-ix by J.Snyder.
23. **Why do we need a map projection?** If you are
mapping a significant portion of the Earth's surface it is impossible
to project it on a flat piece of paper without scale distortions. Map
projections take care that the scale distortions remain within certain
limits and the distortion pattern of a map projection determines the property
of the projection. Each projection has its own characteristics. For example
a map projection may have the property that all angles are correctly represented
(conformal projection property). A map projection is not of major importance
for city or street maps, which cover a relatively small surface of the
Earth.
24. **How do we classify map projections? **Map projections can be classified in terms of their class (cylindrical, conical, azimuthal), their property (equivalent, equidistant,
conformal), their aspect (normal/polar,
transverse, oblique), and its tangent or secant map surface. An example would be the classification ‘conformal conic projection with two standard parallels’ having the meaning that the projection is a conformal map projection, that the intermediate surface is a cone, and that the cone intersects the ellipsoid (or sphere) along two parallels; i.e. the cone is secant and the cone’s symmetry axis is parallel to the rotation axis. Other examples are:
- Polar stereographic azimuthal projection with
secant projection plane
- Lambert conformal conic projection with two standard
parallels
- Lambert cylindrical equal-area projection with
equidistant equator
- Transverse Mercator projection with secant projection
plane.
25. **Why are some map projections using mapping zones?**
There are map projections that divide the mapping area into zones in order to keep
scale distortions within acceptable limits. A good example is the Universal Transverse Mercator (UTM) projection. Each zoning systems has its own map coordinate system (*x*,*y*) with a specific origin. Coordinate transformations are required to match areas that are located in different zones.
26. **How to determine the zon(s) of the UTM for a certain country?** You need to know the east-west extent of the country and compare it to the UTM zone numbering system. The zone numbering system of the UTM projection specifies 60 zones of 6 degrees longitude each. The zones are numbered from 1 to 60 starting with zone number 1 at -180^{o} (West of Greenwich) until -174^{o} (West of Greenwich) and so on. An example, the Netherlands is located between the longitudes 3 and 7 degrees East of Greenwich, so it covers 2 UTM zones, zone number 31 and zone number 32. Zone number 31 covers the area from 0^{o} (Greenwich) until 6^{o} (East of Greenwich), and zone 32 the area between 6-12^{o} E.
27. **How do we match adjacent maps?** In order to fit
two or more separate maps exactly along their edges, a number of parameters
must be maintained: 1. the maps must be constructed with the same projection
and projection parameters; 2. they must be at the same scale; and 3. they
should be based on the same reference datum.
28. **How to select a suitable map projection?** The choice of a map projection class largely depends on the size and shape of
the geographical area to be mapped; cylindrical projections are often used for large
rectangular areas (and to map the world); conic projections for medium size triangular areas (and to map the different continents);
azimuthal projections small-size circular areas (and to map the poles). The choice of a map
projection property has to be made on the basis of the purpose of map; conformal projections are often used for sea, air and meteorological charts, topographic
and large scale maps; equidistant projections are often used for air-route, radio or seismic maps but also for topographic and large
scale maps; equal-area projections are often used for distribution maps and also for historical, population, geological
and soil maps (more information in section 4).
29. **What type of information concerning the map coordinate system should be shown on a map?** Most important items are the scale (graphic and/or numerical), the map projection (incl. ellipsoid and datum), the grid and/or the graticule, and in the border area the grid and graticule values. Other items might be necessary to put a map (e.g. topographic map) in its full use: items such as the vertical (height) datum, unit of elevation (meters/feet), contour interval, graphic scales in kilometers/statute miles/nautical miles, projection details (parameters such as the false easting/northing, central meridian, etc), geographic coordinates of sheet corners, instructions on the use of the grid, information on true, grid and magnetic north, etc. Examples of coordinate information on a topographic map:
Notes concerning the map coordinate system in the legend of a German
topographic map (scale 1:25 000).
Notes concerning the map coordinate system in the legend of maps
distributed by the NIMA.
30. **What are True North, Magnetic North, and Grid
North?** The *True North * is the direction of the meridian to the
North Pole at any point on the map. The *Grid North* is the northern direction of the north-south grid lines
on a map. The *Magnetic North* is the direction
of the Magnetic North Pole as shown on a compass free from error or disturbance. The corresponding
bearings are called: true bearing or geodetic bearing, grid bearing and
magnetic or compass bearing.
Three different angles are in use: The *Magnetic Declination* is the angle between Magnetic
North and True North at any point (in the figure 7 degrees and 36 minutes at the sheet centre of a map). The *Grid Convergence* is the angle between Grid
North and True North. The *Grid Magnetic *Angle is the angle between
Grid North and Magnetic North (in the figure 4 degrees and 58 minutes at the sheet centre of a map). This is the angle required for conversion
of grid bearings to magnetic bearings or vice versa. The *Annual Magnetic Change* (*8 minutes East *in the figure above) is the amount by which
the magnetic declination changes annually because of the change in position
of the magnetic north pole
8.5 FAQ on coordinate
transformations
31. **What is a coordinate transformation? **A
coordinate transformation is a conversion of coordinates from one to another
coordinate system. Transformations can be between plane coordinate systems,
between geographic and plane coordinate systems, between geographic coordinates
and geocentric coordinate systems, etc. See section 5 for details.
32. **What is a map projection change? **The
transformation of coordinates from a plane system based on one projection
type into a plane system based on another type. An example, a projection change can transform *x, y* coordinates
from the UTM coordinate system into the
Lambert Conformal Conical projection system. Forward and inverse mapping equations (see section 4 on map projections) are normally used to transform data from one map projection to another.
33. How to convert a data set from one UTM zone to another UTM zone? Make use of a projection change (see section 5.1). Project the data onto the reference surface (geodetic datum) with the inverse equations of the UTM projection and thereafter project the data onto the target (adjacent) UTM zone with the forward UTM projection equations. The conversion into another UTM zone may also be applied via a 2D Cartesian transformation, but it requires a set of control points (common points) to determine the relationship between the two UTM zones, and the transformation will give a different (perhaps unacceptable) accuracy.
34. **H****ow do you assign coordinates to a data
set with unknown coordinates? **You can use
control points, such as the corners of houses, or road intersections,
with known coordinates taken from a map or another data source to determine the relationship between the unknown and a known coordinate
system. The transformation may be conformal, affine or polynomial depending
on the systematic errors in the data set.
35. **How do you georeference a map (e.g. for digitizing), if the only given coordinate system are geographic coordinates?** Select four graticule corner points and convert these points from geographic coordinates into map coordinates (*x,y*) using the forward equations of the map projection. Directly using the geographic coordinates for the georeferencing may cause geometric and alignment errors.
36. **Why do we need datum transformations?** Numerous
maps are projected onto various ellipsoids or reference datums. Very often,
it is needed to transform one datum to another to avoid alignment problems. See section 5.2 for details. |