The easiest coordinate system to construct is one that we can use to describe the location of objects in one dimensional space. For example, we may wish to describe the location of a train along a straight section of track that runs in the East-West direction.
For example, the origin for our train track may be the Kingston train station Figure A1. If the object is not at the origin, then we need to be able to specify on which side East or West in our train example of the origin the object is located. Thus, in order to fully specify a one-dimensional coordinate system we need to choose:. To describe the position of an object in two dimensions e. These points are used to define the orientation of the axes of the local system; they are the tips of three unit vectors aligned with those axes.
To read the coordinate system you have to know what side is "n" the bottom side with numbers then you go from "n" to whatever your number is. A coordinate transformation is a conversion from one system to another, to describe the same space. For example, in one dimension, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more.
A geographic coordinate system GCS defines locations on the earth using a three-dimensional spherical surface [4] , it is a reference system that uses latitude and longitude to identify locations on a spheroid or sphere. A GCS includes a datum based on a spheroid , prime meridian, and an angular unit of measure.
Coordinates can be described using different frames of reference, or datums that are designed to be more accurate for different areas of the earth. Geographic coordinate systems vary from projected coordinate systems in that they reference the earth as a 3D object measured in degrees rather than using a 2D projection of the earth's surface to measure it using meters or feet.
However, latitude and longitude degrees do not have a standard length. This makes it difficult to measure distance and represent data accurately on a computer screen or flat map.
An example of this would be showing the location of Salt Lake City, UT with the following coordinate pair: Rather than using degrees to represent points on a spheroid, some coordinate systems are placed on 2-dimensional projections and function much like the previously mentioned Cartesian System.
These coordinate systems often use measurements in feet or meters to express x and y coordinates for specific points. The advantages of these systems are in the easy to express and understand coordinates and simpler calculations of distance and area due to constant lengths and angles across the projection. The SPCS is made of zones which provides for high accuracy within each zone but low accuracy outside the zone.
Coordinate system From wiki. Furthermore, the ellipses reveal that the character of distortion associated with this projection is that shapes of features as they appear on a globe are preserved while their relative sizes are distorted.
Map projections that preserve shape by sacrificing the fidelity of sizes are called conformal projections. The plane coordinate systems used most widely in the U. The Transverse Mercator projection illustrated above minimizes distortion within UTM zone 30 by putting that zone at the center of the projection. Fifty-nine variations on this projection are used to minimize distortion in the other 59 UTM zones. In every case, distortion is no greater than 1 part in 1, One disadvantage of the UTM system is that multiple coordinate systems must be used to account for large entities.
The fact that there are many narrow UTM zones can lead to confusion. For example, the city of Philadelphia, Pennsylvania is east of the city of Pittsburgh. If you compare the Eastings of centroids representing the two cities, however, Philadelphia's Easting about , meters is less than Pittsburgh's about , meters.
Because although the cities are both located in the U. As it happens, Philadelphia is closer to the origin of its Zone 18 than Pittsburgh is to the origin of its Zone If you were to plot the points representing the two cities on a map, ignoring the fact that the two zones are two distinct coordinate systems, Philadelphia would appear to the west of Pittsburgh.
Inexperienced GIS users make this mistake all the time. Fortunately, GIS software is getting sophisticated enough to recognize and merge different coordinate systems automatically. The UTM system was designed to meet the need for plane coordinates to specify geographic locations globally.
Focusing on just the U. Chief among these were:. As discussed above, plane coordinates specify positions in flat grids. Map projections are needed to transform latitude and longitude coordinates to plane coordinates.
The designers of the SPCS did two things to minimize the inevitable distortion associated with projecting the Earth onto a flat surface. First, they divided the U. Second, they used slightly different map projection formulae for each zone, one that minimizes distortion along either the east-west or north-south line depending on the orientation of the zone.
The curved, dashed red lines in the illustration below represent the two standard lines that pass through each zone. Standard lines indicate where a map projection has zero area or shape distortion some projections have only one standard line.
As shown below, some states are covered with a single zone while others are divided into multiple zones. Each zone is based upon a unique map projection that minimizes distortion in that zone to 1 part in 10, or better. In other words, a distance measurement of 10, meters will be at worst one meter off not including instrument error, human error, etc. The error rate varies across each zone, from zero along the projection's standard lines to the maximum at points farthest from the standard lines.
Errors will be much lower than the maximum at most locations within a given SPC zone. SPC zones achieve better accuracy than UTM zones because they cover smaller areas, and so are less susceptible to projection-related distortion. As we have seen above, positions in any coordinate system are specified relative to an origin. Like UTM zones, SPC zone origins are defined so as to ensure that every easting and northing in every zone are positive numbers.
As shown in the illustration below, SPC origins are positioned south of the counties included in each zone. The origins coincide with the central meridian of the map projection upon which each zone is based. The false origin of the Pennsylvania North zone, is defined as , meters East, 0 meters North. Origin eastings vary from zone to zone from , to 8,, meters East. The starting "0" for states in the range is typically dropped; thus for CA, as an example, the most norther SPCS zone is Shown below is the southwest corner of the same ,scale topographic map used as an example above.
The tick labeled "1 FEET" represents a SPC grid line that runs perpendicular to the equator and 1,, feet east of the origin of the Pennsylvania North zone. Notice that, in this example, SPC system coordinates are specified in feet rather than meters.
The SPC system switched to use of meters in , but most existing topographic maps are older than that and still give the specification in feet as in the example below. The origin lies far to the west of this map sheet. Unlike longitude lines, SPC eastings and northings are straight and do not converge upon the Earth's poles. SPCs, like all plane coordinate systems, pretend the world is flat. The basic design problem that confronted the geodesists who designed the State Plane Coordinate System was to establish coordinate system zones that were small enough to minimize distortion to an acceptable level, but large enough to be useful.
Most SPC zones are based on either a Transverse Mercator or Lambert Conic Conformal map projection whose parameters such as standard line s and central meridians are optimized for each particular zone. One of Alaska's zones is based upon an "oblique" variant of the Mercator projection.
That means that instead of standard lines parallel to a central meridian, as in the transverse case, the Oblique Mercator runs two standard lines that are tilted so as to minimize distortion along the Alaskan panhandle. These two types of map projections share the property of conformality , which means that angles plotted in the coordinate system are equal to angles measured on the surface of the Earth.
As you can imagine, conformality is a useful property for land surveyors, who make their livings measuring angles. This section has hinted at some of the characteristics of map projections and how they are used to relate plane coordinates to the globe.
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